The iteration of cubic polynomials Part II: patterns and...

Acta Math., 169:3--4 (1992), 229-325

The iteration of cubic polynomials

Part II: Patterns and parapatterns

BODIL B R A N N E R

The Technical University o f Denmark Lyngby, Denmark

b y

and J O H N H. H U B B A R D

CorneU University Ithaca, N.Y., U.S.A.

CHAPTER 1.1. 1.2.

CHAPTER 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

CI~AZI'ER 3.1. 3.2.

CHAPTER 4.1. 4.2. 4.3. 4.4. 4.5.

CHAPTER 5.1. 5.2. 5.3. 5.4.

CHAPTER 6.1. 6.2.

Table of contents

I. IN~ODUC'nON ......................... 230 Patterns and their origin in the dynamical plane ........ 231 Outline of the paper ....................... 233

2. PAYrERNS ........................... 235 Topological preliminaries .................... 235 C o n s t r u c t i n g the t ree o f p a t t e r n s . . . . . . . . . . . . . . . . 237 T he po ten t i a l func t ion hR . . . . . . . . . . . . . . . . . . . . 239 Cri t ical g raphs , annul i and a r g u m e n t s . . . . . . . . . . . . . 240 T h e t ree o f real p a t t e r n s . . . . . . . . . . . . . . . . . . . . 242 P a t t e r n s o f inf ini te d e p t h . . . . . . . . . . . . . . . . . . . . 245 P a t t e r n i s o m o r p h i s m s . . . . . . . . . . . . . . . . . . . . . . 245 The p a t t e r n b u n d l e . . . . . . . . . . . . . . . . . . . . . . . 249 The quo t i en t p a t t e r n b u n d l e . . . . . . . . . . . . . . . . . . 251

3. PAI'FERNS AND POLYNOMIALS . . . . . . . . . . . . . . . . . 251 Po lynomia l s rea l iz ing p a t t e r n s . . . . . . . . . . . . . . . . . 252 I n t r oduc i ng p a r a m e t e r s in T h e o r e m 3.1 . . . . . . . . . . . . 255

4. ENDS OF PATTERNS . . . . . . . . . . . . . . . . . . . . . . 257 P a t t e r n s and t he i r ends . . . . . . . . . . . . . . . . . . . . . 257 N e s t s and t ab l eaux . . . . . . . . . . . . . . . . . . . . . . . 258 P rope r t i e s of t ab l eaux . . . . . . . . . . . . . . . . . . . . . . 259 T h e rea l izabi l i ty of t ab l eaux . . . . . . . . . . . . . . . . . . 261 Modul i of nes t s . . . . . . . . . . . . . . . . . . . . . . . . . 264

5. JULIA SETS OF CUmCS . . . . . . . . . . . . . . . . . . . . . 269 T h e ma in s t a t e m e n t s . . . . . . . . . . . . . . . . . . . . . . 272 Ana ly t i c p re l iminar ies . . . . . . . . . . . . . . . . . . . . . . 273 Proofs of the t h e o r e m s . . . . . . . . . . . . . . . . . . . . . 274 T he m e a s u r e o f Jul ia sets . . . . . . . . . . . . . . . . . . . . 278

6. THE MONODROMY OF PATTERNS . . . . . . . . . . . . . . . . . . 281 T he local t r iv ia l iza t ion o f the p a t t e r n b u n d l e . . . . . . . . . . 281 T h e m o n o d r o m y o f the p a t t e r n bund le . . . . . . . . . . . . . 283

230 B. BRANNER A N D J. H . HUBBARD

6.3. The monodromy of the quotient pattern bundle . . . . . . . . 285 6.4. Fractional Dehn twists . . . . . . . . . . . . . . . . . . . . . 286

CHAPTER 7. PARAPATTERNS . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.1. Parapatterns for fixed ~ . . . . . . . . . . . . . . . . . . . . . 288 7.2. The potential function H, critical graphs and arguments . . . 289 7.3. The real parapattern . . . . . . . . . . . . . . . . . . . . . . 289 7.4. Parapattern isomorphisms . . . . . . . . . . . . . . . . . . . 290 7.5. The parapattern bundle . . . . . . . . . . . . . . . . . . . . . 291 7.6. The quotient parapattern bundle . . . . . . . . . . . . . . . . 292 7.7. The bundle of clover leaves . . . . . . . . . . . . . . . . . . . 292

CHAPTER 8. PARAPATrERNS AND POLYNOMIALS . . . . . . . . . . . . . . . 292 8.1. The universality of parapatterns . . . . . . . . . . . . . . . . 293 8.2. Parapatterns and the escape locus . . . . . . . . . . . . . . . 293

CHAPTER 9. ENDS OF PARAPATI~RNS . . . . . . . . . . . . . . . . . . . . 296 9.1. The two types of ends . . . . . . . . . . . . . . . . . . . . . 296 9.2. Mandelbrot-like families in cubic polynomials . . . . . . . . . 297

CHAPTER 10. T H E MONODROMY OF PARAPATrERNS . . . . . . . . . . . . . 301 10.1. The local trivialization of the parapattern bundle . . . . . . . 302 10.2. The monodromy of the parapattern bundle . . . . . . . . . . 303 10.3. 10.4. 10.5.

CHAPTER 11.1. 11.2. 11.3. 11.4. l l .5 . 11.6.

CrIAPTER 12.1. 12.2. 12.3. 12.4. 12.5.

The monodromy of the quotient parapattern bundle . . . . . . 303 Fractional Dehn twists . . . . . . . . . . . . . . . . . . . . . 304 Consequences for the polynomials of infinite depth . . . . . . 304

11. T H E FUNDAMENTAL GROUP OF THE ESCAPE LOCUS . . . . . . . 309 Choice of base point . . . . . . . . . . . . . . . . . . . . . . 310 The fundamental group of the real parapattern . . . . . . . . 311 The fundamental group of the parapattern bundle . . . . . . . 312 Computing the monodromy of the parapattern bundle . . . . . 314 The fundamental group of the quotient parapattern bundle . . 315 The fundamental group of the escape locus . . . . . . . . . . 317

12. POLYNOMIALS OF mGHER DEGREE . . . . . . . . . . . . . . . 319 The global topology of parameter spaces . . . . . . . . . . . 320 Patterns of higher degree . . . . . . . . . . . . . . . . . . . . 321 Ends of patterns and tableaux . . . . . . . . . . . . . . . . . 321 Julia sets of polynomials of higher degree . . . . . . . . . . . 323 Parapatterns in higher degree . . . . . . . . . . . . . . . . . . 323

12.6. Ends of parapatterns and Julia sets . . . . . . . . . . . . . . 324 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Chapter 1. Introduction

This is the s e c o n d in a ser ies o f 3 paper s devo ted to the desc r ip t ion o f c o m p l e x cub ic

po lynomia l s c o n s i d e r e d as d y n a m i c a l sys t ems .

F r o m F a t o u [F], Ju l ia [J] and M a r i e - S a d - S u l l i v a n [MSS] , we k n o w that a c o a r s e

c lass i f ica t ion is g iven by s imply coun t ing the n u m b e r o f cr i t ical po in t s wh ich e s c a p e to

INTRODUCTION 231

infinity. In the previous paper [BH], we identified the topology of the locus of polynomials

ea, b(z) = Z 3-3a2z+b

where at least one critical point escapes. In this paper we will describe how that locus is broken up according to whether one or both critical points escape.

This paper and the previous one give a complete description of the space of cubic polynomials for which one critical point escapes. This is of interest in itself, but should also provide tools for creeping up to the cubic connectedness locus: the locus where neither critical point escapes. More particularly, it should allow us to give a partial description of this locus in terms of stretching rays; this will be the object of the third paper.

1.1. Patterns and their origin in the dynamical plane

As usual in complex analytic dynamics, we plough in the dynamical plane and reap in the parameter space. We will analyze carefully the dynamical plane for cubic polynomials P where at least one critical point escapes. We will do this by building "abstract Riemann surfaces", called patterns, which will turn out to be isomorphic to appropriate domains in the dynamical plane. However, precisely because they have been constructed without any reference to dynamical systems, and in particular are independent of any "infinite process", we can understand them more easily than the dynamical domains they represent.

An outline of the procedure is as follows. Let he be the potential defined by

h e (z) = lim 1 log+ le~ I n--,oo 3 n

as in [DH1, BH], and define the filled-in Julia set as

K e= { z E C Ihe(z) =0} .

For any z0 E C let

Ue(z o) = {z E CI he(z) > he(zo} ,

so that Up(zo)=C-Kp if zoEKe. Suppose P has a critical point tol escaping to infinity, i.e. he(tOl)>O, and another

232 B. BRANNER AND J. H. HUBBARD

critical point to2 escaping to infinity more slowly than to~ or not at all, i.e. hp(to2)<he(toO. The domain we wish to reconstruct abstractly is Ue(to2); note that Up(toE)=C-g P i f (,0 2 does not escape. Of course, as a subset of C these sets are complicated, with boundaries given by transcendental equations if hp(to2)>0, and fractal if he (to2)=0. They are difficult to "know" in any very precise sense. Forgetting about their embedding in C and remembering only their structure as Riemann surfaces, they are also fairly complicated, depending on a complex number ~ E C - / ) , a real number h<log ]~1 and many (sometimes infinitely many) combinatorial data. But for all that they are "knowable", as follows.

The first step is to construct an analytic mapping q0~, defined in a neighborhood of infinity and conjugating P to z,--~z 3. There are only two such q0e, and if P is monic there is a unique one which is tangent to the identity at o~. Now for purely topological reasons, q0e can be extended to Ue (wO, and qge (Ue (to l ) ) = C - D r where D r is the disc of radius r=e her'~ Thus, as an abstract Riemann surface, Ue(tOl) is simply the complement of a disc. The point P(to0 is in Ue(w~), and the number q~e(P(wl)) is a dynamical invariant of the polynomial. Call it ~3, because we will see in a moment that it has a distinguished cube root.

Observe that Ue(tol) is a domain in C bounded by a curve homeomorphic to a figure eight, with double point the critical point to~; and that this curve contains one extra inverse image of P(to0, which we will call the co-critical point to~'. The mapping qoe can be extended to a neighborhood of Wl', and q~e (~01')=r is the distinguished cube root mentioned above.

From the complex number r we can reconstruct Ue(to~) as C-DIe I with the points jr and j2r identified, where j and j2 are the non-real cube roots of 1.

Now consider P-l(up(a~l)); if hp(w2)<~hl,(tOl), the mapping P:P-l(Ue(tol))--~ Ue(to ~) is a ramified triple covering, ramified at the single point tot. There are exactly two such coverings, classified by the component of C-Ue( to 1) containing the critical value P(t09.

If he(~o2)<~he(tol), the next inverse image P-2(Ue(to O) is again specified by the component of C-P-~(Ue(to~)) containing P(to2); of course the enumeration of the possible components depends on our previous choice.

This construction of successive inverse images of Ue(to l) continues until an inverse image contains P(t02), and can be continued for ever if to2 EKe.

The successive specification of the component of C-P-n(Ue (to 1)) containing P(t02) is called the combinatorial information. With ~ E C - / ) and h<log fixed and he (to2)<h the structure of the set of values of z such that he(z)>h only depends on the combina-

INTRODUCTION 233

torial information; it remains isomorphic to itself as a Riemann surface when P ranges in a component of the set of polynomials defined by q~e(~ol')=~ and he (to2)<h.

1.2. Outline of the paper

A pattern is a sequence of recursively chosen covering spaces as above. The only analytic information is the number ~, the remainder of the information is combinatorial, specifying covering spaces in such a way as to capture the "essence" of cubic polynomials. In Chapter 2 of this paper we describe exactly how to build them. It is fairly easy to show that the domains in the dynamical plane described above are isomorphic as Riemann surfaces to appropriate patterns, and this is done in Chapter 3. It is much less obvious that all patterns can be realized in this way.

Patterns come with embedded graphs which divide them up into annuli. The combinatorial description of the covers turns out also to encode the moduli of the annuli, and we are able under appropriate circumstances to estimate these moduli so as to show that components of Ke are points. This, together with the theory of polynomial- like mappings [DH2], leads to quite a precise description of the Julia sets of cubic polynomials for which one critical point escapes. In particular, we give a complete characterization of those polynomials for which the Julia set is a Cantor set. Theorems 5.2 and 5.3 give the precise statements, but the hard work is done in Chapter 4, where a tableau is associated to any pattern; it contains enough information to make the estimates on annuli referred to above, but sufficiently little to be reasonably manage- able. There the main result is Theorem 4.3.

Curt McMullen proved that those Julia sets to which the tableau argument applies (these are precisely those which are Cantor sets) are of measure zero, and has kindly agreed to our including his result as Theorem 5.9 of our paper.

Next we take up the problem of showing which patterns actually occur. The result is very satisfying: Theorems 8.2 and 9.1 say that every pattern does occur, and tells exactly how two polynomials are related if they have the same pattern.

We could have proved the results of Chapter 5 without ever mentioning patterns, simply by working in the dynamical plane itself. But in Chapters 7, 8 and 9 we see what the drudgery in Chapters 2 and 3 has bought us: a complete abstract description of the escape locus: the space of cubic polynomials for which both critical points escape. In Chapter 7 we set up a parameter space for patterns which is itself a complex manifold, called the parapattern space. It is universal in the sense that it parametrizes patterns,

16-928286 Acta Mathematica 169. Imprim6 le 10 novembre 1992

234 B. B R A N N E R A N D J . H . H U B B A R D

and there is a map from the escape locus into it, classifying the patterns of polynomials. Our existence and uniqueness statement says that this mapping is an isomorphism: the proof consists essentially in showing that this mapping is analytic and proper.

Further we are able to understand the structure of the locus ~ where one critical point escapes and the other does not. It is a fibration over C - / ) . The fiber is made up of uncountably many components, countably many of which are homeomorphic to the Mandelbrot set, i.e. the connectedness locus for quadratic polynomials, and the others are points. The first fact is an application of Mandelbrot-like families of mappings [DH2] and the latter fact (like Theorem 5.2) an application of the tableau argument. Hence we have shown that there are no "queer" components in the fiber. This result can be viewed as an analogue of the following conjecture: the natural map from the Mandelbrot set onto the abstract Mandelbrot set (defined by Thurston) is a bijection; i.e. there are no "queer" inverse images of points. This conjecture is equivalent to the local connectivity conjecture of the Mandelbrot set.

Chapters 10 and 11 are devoted to understanding the topology of the parapattern space. The escape locus, or rather the subset where the critical point +a escapes faster than the other critical point - a , is also topologically a fiber bundle over C- /5 , and in Chapter 10 we compute its monodromy. This allows us, at least in principle, to understand the components of the locus where one critical point escapes and the other doesn't. It also allows us to compute the fundamental group of the escape locus, in Chapter 11. The main motivation for doing so is a recent result of Blanchard, Devaney, and Keen [BDK] showing that there is a representation of this fundamental group onto the group of automorphisms of the one-sided 3-shift.

We would like to call the attention of the reader to Chapter 12. This chapter says that most of the constructions of this paper and the previous one go over to polynomials with two critical points of arbitrary degree. But the analogue of Theorem 5.2 on Julia sets does not go through, and gives a place to look for possible counterexamples to the generic hyperbolicity conjecture.

Acknowledgements. This paper was almost five years in the writing, and went through many drafts. While (and before) this paper has been written several other papers [Bll], [Bl2], [BDK], [Br], [L], [M2], [M3] containing studies of cubic polynomials have appeared (at least as preprints) and we have benefited from reading them. In particular we want to emphasize that Pierre Lavaurs [L] settled the conjecture in [BH], showing that the connectedness locus is cellular in all degrees.

During these five years the authors benefited from help, conversations, sugges-

PAT'FERNS 235

tions and criticism from many people, and hospitality and funding from many institutions. We wish to thank all of them here.

As always in this field, Adrien Douady must be mentioned first: he provided innumerable comments and suggestions without which this paper would be very different, or perhaps would not exist.

Very special thanks are also due to Homer Smith. t ie made the computer pictures which appear in the paper, and also wrote a number of programs for exploring cubic polynomials, without which the results might never have been found.

Curt McMullen found the proof of Theorem 5.9 and has been so kind as to allow us to include it in our paper. Mitsuhiro Shishikura pointed out that Theorem 4.3(a) failed in higher degrees, correcting an error in an earlier draft of the paper.

We do not know exactly who had the idea of using moduli of annuli to show that some Julia sets are Cantor sets, but it wasn't one of us. We believe that the credit should be shared by Adrien Douady and Paul Blanchard.

We thank Ben Bielefeld, Robert Devaney, Yuval Fisher, Lisa Goldberg, Linda Keen, Pierre Lavaurs, John Milnor, Mary Rees, Dennis Sullivan, Tan Lei and Bill Thurston, for many helpful conversations.

The authors thank the Max Planck Institut for Mathematik, the Humboldt Founda- tion, the Mittag-Leffler Institute and N.S.F. Grant DMS 86-01016, as well as our respective home institutions, Cornell University and the Technical University of Denmark, for the material support without which no scientific research can take place.

Chapter 2. Patterns

In this chapter we shall build for each r E C - / ) a tree

~C) = U ~.(~) n

of "abstract" Riemann surfaces. This is not very hard once one has the right defini- tions; none of the proofs are difficult. But it is quite long and cumbersome, especially as the reason for the construction must wait till Chapter 3. These Riemann surfaces are combinatorial reconstructions of domains which arise naturally as subsets of the dynamical plane.

2.1. Topological preliminaries As indicated in the introduction, our patterns are built from towers of coveting spaces. In Section 2.8, we will make bundles out of our patterns; we must therefore think


functorially, i.e. construct objects not just up to isomorphism but up to unique isomorphism. The following two lemmas are topological existence and uniqueness statements; the uniqueness part will allow just that.

LEMMA 2.1. Let f : S1---~S 1 be a homeomorphism; then f extends to a homeomorphism F: 19---~1) such that F(0)=0, and any two such extensions are isotopic among such homeomorphisms.

Proof. The mapping f can be extended radially. The isotopy statement is proved using the Alexander trick. If F1 and F2 are two extensions, set F=F1 oF2 -l. Then the family

x i f Ilxll > t F(t, x) = tF(x/t) if Ilxll ~< t

provides an isotopy of F to the identity. Q.E.D. of Lemma 2.1

LEMMA 2.2. Let a and b be two points in X=C. Then there exists a connected triple cooer Jr: Y--~X ramified abooe a and b such that ~r-l(a) and ~r-l(b) both consist o f precisely two points, and i f ~rr: Y--~X and ~rz: Z---~X are two such cooers, there exists a unique cooering homeomorphism Y---~Z.

Figure 2.1 illustrates such a ramified covering, as well as the notation used in the proof.

Y

a b Fig. 2.1

PATTERNS 237

Proof. The cubic polynomial

= b-a(z3_3z) + a+b P(z) 4 2

has the property that P: C--->C is a connected triple cover ramified above a and b with P-l(a) = ( 1,-2} and P-l(b) = { - 1,2}. This establishes existence.

The uniqueness and rigidity can be seen as follows. Let Do, DbcX be closed discs containing a and b respectively in their interior, and with Do n Db = {x} a single point. Let :ty: Y-->X and :tz: Z--->X be two ramified triple covers as in 2.2. Define Ya=:ry-l(Da), etc. Our hypotheses imply that each of Ya, Yb, Zo, Zb consists of two discs (Yo)I, (Y~)2, etc. such that the first covers its image with degree 1 and the second with degree 2. The set (Yo)2N(Yb) 2 consists of a single point y; indeed it cannot be empty because both Yan :rr-l(x) and Yb n : t r- l(x) consists of two points, and :rr-l(X) only has three points; and it cannot contain two points for otherwise Y would be disconnected. Similarly (Za)2 n (Z~)2----- {z}.

There now exists a unique covering homeomorphism (Y~)I---~(Z~)I since these are simple covers, and a unique covering homeomorphism (u mapping y to z; and similarly with a replaced by b. These four homeomorphisms piece together to give a covering homeomorphism of ~rr-l(DaUDb) to ~tz-l(D~UDb). Since the inclusion of DaUDb into X is a homotopy equivalence, this homeomorphism extends uniquely to a covering homeomorphism Y---~Z.

The rigidity follows also: any covering transformation has to map y to z, so the covering homeomorphism is unique. Q.E.D. for Lemma 2.2

Definition 2.3. The triple cover described byL emma 2.2 will be called the standard triple cover.

2.2. Constructing the tree of patterns

Choose a complex number ~ with I~l=r>l. We shall recursively construct for each N E Z

(a) a finite set ~N(~) of Riemann surfaces, called E-patterns of depth N, and (b) a mapping SN: ~N(~)--->~N_I(~).

Each R E ~N (~) will come with (c) an embedding into a simply connected Riemann surface/~ isomorphic to C; (d) if N > 0 a distinguished point t0R E R; (e) a triple covering map :tR:R-->SN(R) ramified at wR (i.e. unramified if N~<0);


(f) an inclusion in: SN(R)-->R, satisfying i~(n ) o ~LN~R)=~rR o iR; and (g) a collection of open annuli Bn(R)~R, n<N, nested in/~, such that

i n (B,(s N (R))) = B,(R) for n < N - I.

Construction o f ~N(~), N~<0. The set ~N(~) consists for each N~<0 of the single Riemann surface RN=C-19yN , and/~N=C. All maps ~rRN: RN---,RN_ 1 are given by z ~ z 3, and the mappings inN: RN_I--->R N are the inclusions. Finally set

B~(RN) = {zl r3-" < Izl < r3-'§ Construction o f ~1(~). Again ~1(~) consists of just one Riemann surface RI. Using

Lemma 2.2, there exists a unique standard triple cover

of C=/~0 ramified above ~3 and 0; let R 1 =(:rR,)-I(Ro) and :rR: Rr 0 be the restriction of ~rR. Further let toR~ be the critical inverse image of r and to'R~ be the other (co- critical) inverse image. This specifies parts (a)-(e) of the construction, for n= 1. Note that so far we have only used r and not ~.

For part (f), observe that the mapping z ~ z 3 makes Ro into a connected triple cover of R_~, and

- - the restriction of ~rRI: RI---~R 0 to (:rR~)-I(R_I) is also a connected triple cover of R - 1 .

Therefore, there are three covering isomorphisms

j: R0--* (zrR,)-I(R_I).

Precisely one of these satisfies limz_~r This is the mapping iRl: Ro--.R 1. Define Bo(R1)=iR~({z I r<lzl<?}). See Figure 2.2.

Construction o f ~N(~),N>I. The elements of -1 SN+1(R) will be indexed by the bounded components of R-BN_I(R) . We assume by induction that each such component A is an annulus which shares a boundary with a unique component Da o f / ~ - R , homeomorphic to a disc. Pick a point XEDA. Let ~rRA:/~a--~/~ be the standard triple cover of/~ ramified above x and in(arn(~OR)); let RA=(~rRA)-l(R) and ~rnA be the restriction of er,~ to Ra. (Figure 2.3 shows construction of ~2(~) from ~1(r

We still need to construct the inclusion in: R - . R a. The surface R was constructed from SN(R) using the choice of a point yE SN(R); extend iR: SN(R)--,R to a homeomor-

R1 O)R~

PATTERNS

Fig. 2.2

239

phism/-R: SN(R)---~R so that ~R(y)=x. Such an extension exists by Lemma 2.1, and all such extensions are isotopic.

By the functoriality of the triple cover (Lemma 2.2) the mapping IR lifts uniquely to a homeomorphism/~---~/~a. The inclusion ira is the restriction of this lifting to R. It is clearly independent of the extension/-R chosen. The annulus BN(RA) is iRa(A).

Remark 2.4 (a) The construction of ~N(~) for N = 1 is essentially a special case of the construction for N > 1. Only the definition of iR was different.

(b) The construction of/~A from R depends not only on A but also on the choice of a point x EDA. It is easy to check, however, that neither R a, ~rRA, or iRA depends on this choice (up to unique isomorphism of Riemann surfaces).

2.3. The potential function hR

For any R E ~N(~), there is a unique harmonic function hR:R-->R+ which "extends" the function log Izl. It is defined recursively on patterns of increasing depth by setting


R 2 q

(DR 2

RI I ~ . ~ l ~ .~?~tiRI(~7"~RI((])~I ))

The s tandard triple cover ramified above iR~(:rR~(ton~)) and x~ E D~,x2 E D2 respectively. The arrow is drawn from an annulus of R to an annulus o f R~=s2(R). The annulus o f R is the unique annulus at level 2 which double covers an annulus o f R~.

Fig. 2.3

hn(z) = 3 h sNcm(:rn(z) )'

and hn(z)=log [zl when N~<0. This function satisfies hR(in(z))=hsN<m(z). Use this function to define subsets

f log Rn = ~z E R hn(Z) >

3" )

we see from the two formulas above that in induces an isomorphism of the pattern SN(R) with Rn_ 1. Thus each Rn is canonically isomorphic to a ~-pattern in ~ (~ ) , allowing us to think of a pattern as an increasing union of Riemann surfaces with appropriate maps; we will f requently allow ourselves to speak of sn(R) as a subset of R.

2.4. Critical graphs, annuli and arguments

The critical values of hs are the numbers (logr)/3 n, for 0~<n<N, where r=l~l. Let

{ l~ F~(R) = z E R hn(z) = 3-T2i-_~ J for n ~ N;

PATTERNS 241

for n~<0, Fn(R) is a simple closed curve, but for n>0, F~(R) are graphs with double points. Further set

F(R) = I.I F,(R). n<~N

The number of critical points of hR belonging to F~(R) is 3 "-~, since there is one critical point COR on Fl(R) and for n~>2 a point x E Fn(R) is critical if and only if erR(x) E F,_I(R) is critical.

Define the annuli of R to be the connected components of R-F(R). The annuli in R,-R,_~ are said to be at level n. At each positive level n there is exactly one annulus Cn(R) which is mapped by err with degree 2, and its image is Bn_~(R); all other annuli of positive level are mapped by err with degree 1 (cf. Figure 2.3). The annulus C,(R) is called the critical annulus of R at level n, and B,_I(R ) the critical value annulus of R at level n - 1.

The mapping JrR: F,+~(R)---~F,(R) is a triple cover for any n> l . One component F'(R) of F,(R) is special, the critical value graph at level n, the one surrounded by the critical value annulus B,_I(R ). The inverse image er~l(F'n(R)) has two components, while for all other components A of F,(R) the inverse image ern-~(A) has three components. Since FI(R) is a figure eight and therefore has just one component, it is easy to find the number of components of F,(R) to be (1+3"-1)/2 for n~>l (see Figures 2.3 and 2.5).

For any R E ~n(~), the critical graph F0(R) is always the same: it is the circle of radius l~l 3. Therefore we can call it simply F0, and use it to compare arguments of points even if they lie in different patterns of ~,(r The argument arg(z) of a point z E F0 is 0 E R/Z where z = rae z~i~ The set of arguments arg(z)cR/Z of a point z E R is the set of arguments of points of F0 at which the (possibly broken) ascending rays from z meet F0. There is a unique argument if the ascending ray contains no critical point of hR. If the ascending ray from z meets such a critical point x then there are several ways of continuing it with ascending rays from x. The argument of the intersection of all such ascending rays with F0 are then the arguments of z.

For any component A of Fn(R), define the arguments of A,

arg(A) = t.I arg(z) c R/Z. zEA

Clearly arg(A) is for each A some finite union of closed intervals, and these form a cover of R/Z and are disjoint except at their endpoints as A runs through all components of Fn(R) for any fixed n~ > I.


The following proposition sounds analogous but isn't really: it is a "parameter statement"; it will be essential in Chapter 11.

PROPOSITION 2.5. For any N ~ I, the sets arg(F 'N(R)) for R E ~N(~) form a covering of the circle R/Z by closed intervals, disjoint except at their endpoints. These endpoints are precisely the arguments o f all critical points of all hR which lie in (F'n(R) for some RE ~(~) and some n with l<~n<N.

Proof. The proof, as always, goes by induction on N. It is clear if N = 1, since in that case there is only one pattern R E ~1(~), and the graph FI(R)=F ~(R) has only one component.

Assume by induction that the result is true for all M<N. For each RE ~N-~(r we have that

LI arg(F'(R')) = arg(F '(R)) s~R')=R

since each annulus inside F '(R) is the critical value annulus B~(R') for precisely one R' E s~vl(R). Q.E.D. for Proposition 2.5

When drawing an element R E ~n(r the main objects represented are the critical graphs Fn(R) together with the gradient curves of hR emanating from the critical points of hR. The insight into the structure of such Riemann surfaces R came to the authors largely by drawing lots and lots of graphs.

2.5. The tree of real patterns

Choose r>I real, and consider the case ~=r.

PROPOSITION 2.6. Suppose R E ~ ( r ) , r> l . Each critical point o f the potential function hR: R-->R+ has precisely two ascending rays, which contain no further critical points and intersect the circle F-I(R) at points with arguments pl/3 k and p2/3k for some integers Pl and P2 not divisible by 3. Moreover all numbers p/3k for p= 1 ..... 3 k - 1 are obtained this way.

Remark 2.7. The statement is true because we are speaking of a pattern in ~(r) with r real. It can fail for other values of ~. For instance, for ~=re 2~/3, a pattern R in ~ ) for N > I looks precisely like Figure 2.4 (a) or (b) down to level 2; in particular ascending rays emanating from some critical points of hR which lie in F2(R) meet the critical point wR.

PATrERNS 243

2/9 2/9

1 / 3 / ~ ' - - ~ " ~ ro

Fo

0 0

2/3 ~ 2/3

(a) (b)

Fig. 2.4

This occurs because the arguments of the ascending rays emanating from wR are 0 and 2/3 (the ray of argument 1/3 leads to the co-critical point). The thirds of these two angles are {0, 1/3, 2/3} and {2/9, 5/9, 8/9} respectively. Two of these thirds coincide with the angles already used. The point of the proof is to show that if we choose ~ real, this phenomenon does not occur, and thirds of an argument leading to a critical point at level N do not coincide with arguments of rays leading to critical points at lower levels.

Figure 2.5 represents ~N(~) for r real and positive, and N = I , 2, 3. The arrows associate to each R E ~u(r) the annulus A of su(R) which labels it, i.e. such that (SN(R))A=R.

Proof of Proposition 2.6. First observe that for each k~> 1 the number of rationals of the form p/3 k with p = 1 .. . . . 3 k - 1 and p and 3 relatively prime is 2.3 k-~.

Moreover, each rational of the form p/3 k with p = 1 ..... 3 k- 1 and (p, 3)= 1 has the following preimages under multiplication by 3 in R/Z:

p p+3 k p+2 .3 k" 3k+j, 3k+l ' 3k+j '

i.e. of the same form.


Rl

)1 Rl

R2

JrR 3

~rR 3

~R 3

R3

zrR 3

An arrow is drawn f rom the critical annulus at level N of RE ??xtr) to the annulus A of sv(R) such that iR(A)=BN_j(R).

Fig. 2.5

PATrERNS 245

Recall from Section 2.4 that the number of critical points of hR belonging to F.(R) is 3 "-~. We will prove the result by induction and start the induction by observing that there are precisely two ascending rays emanating from toR with arguments 1/3 and 2/3.

The inductive hypotheses: The ascending rays emanating from critical points in F.(R) intersect F_~(R) at points of arguments p/3" with p = l ..... 3"-1 and (p, 3)=1. In particular they do not contain the critical value zrR(toR)=r 3 of aiR.

The inverse image of two ascending rays emanating from a critical point of hR in F.(R) consists of 6 ascending rays emanating in pairs from 3 critical points in F.+~(R). From the count above, we see that the rays with arguments p'/3 "+1 with p ' = l ..... 3"+~-1 and (p', 3)=1 account for all such inverse images.

Q.E.D. for Proposition 2.6

2.6. Patterns of infinite depth

A ~-pattern of infinite depth is a sequence (R~)_~<.<| ~ ~ ( ~ ) with each R.E ~.(~) and Sn(R.)=R~_ 1.

We can associate to each infinite pattern the Riemann surface

R| = li_mm(R., iR.+),

and since the critical points correspond under the inclusions, R~ has a distinguished point toR. The mapping erR~: R~--,R| induced by the maps :rR, is a triple covering map ramified at t0R|

The potential functions hR, are also compatible under the inclusions and induce a harmonic function hR: R~---~R+ satisfying hR(Z)=~hR(srR(Z)).

Given the Riemann surface R~ we can again define the subsets

/ Each R" is canonically isomorphic to the ~-pattern R. E ~(~) , so we shall identify them.

2.7. Pattern isomorphisms

Let R and R' be two patterns of depth N ~ < ~. An isomorphism f: R--~R' is a collection of analytic isomorphisms

fn:Rn R., n<.N,

conjugating all ~rR. to ztR~ and iR. to iR~.


This definition is actually much too strong if I<N~<~: all the structure of the pattern up to the determination of + ~ is already encoded in the analytic structure of the Riemann surface R as the Propositions 2.8 and 2.9 will show.

PROPOSITION 2.8. I f g and R' are two patterns o f depth N and N ' respectively, I<N, N'<. o~ and if F: R--~R' is an analytic isomorphism, then N = N ' and the family o f maps f=(fn)n<-N given by fn=FlR" is an isomorphism o f patterns.

Proof. The proof consists of showing that all the pattern structure, except the number ~ which we will deal with below, can be reconstructed from simply the Riemann surface.

If depth(R)=N is finite then the function hR is a harmonic function on R having a logarithmic pole at ~ and such that hR(z)--~(log 1~1)/3 N as Z tends to the other boundary components of R. By the maximum principle, this shows that hR is determined by the analytic structure of R, the choice of the end oo and the number r=[~]. Since the depth of the pattern R is the number of critical values of hR, we see that N is also determined by the data above.

The end ~ is the unique puncture of R, so it depends only on the analytic structure of R.

Reconstructing the number r is a bit trickier. Let g be the harmonic function on R having a logarithmic pole at ~ and tending to 0 on the other boundary components. By the maximum principle for harmonic functions, g=hR+C for some constant C, so hR and g have the same level curves.

If ao>a~>.., are the critical values of g, then

Rn = {z E RI g(z) > a~}.

The region Rl-clos(R0) consists of two annuli At,A2 with moduli Mr,M2 satisfying Ml=2M2, and r is determined by the equation Ml=(logr)/zc.

The mappings iR: R~_I--~R ~ are simply the inclusions. Finally we reconstruct the mappings ZtRn: R~--~R~_ I as follows: the harmonic func-

tion hR has a unique critical point toR at level log r and a unique critical point ~ at level log r/3 in the closure of A I. Set A0=R0- clos(R_ i) where R_ 1 = {z E RNI hR(Z)> 3 log r}. The subset A0 is an annulus with modulus M1 so there exists a unique isomorphism of AI onto A0 mapping ~ to ogR. This mapping must therefore coincide with ~R~ for any l<n<~N. Therefore ersn is defined by analytic continuation.

If depth(R) is infinite then the point ~ is the only isolated end of the Riemann surface and hR, is the unique harmonic function having a logarithmic pole at ~ and

PATTERNS 247

tending to 0 at the other ends. Exactly as for a pattern of finite depth we can construct the number r=loglr the subsets R~, the point ~o,% and the mappings iRo and :rR,.


If R is a C-pattern of depth N with N > 1, then the number r is not determined by the Riemann surface R, but r is. The mapping iR o Z~R is an analytic mapping which has at oo a fixed point which is simultaneously a critical point of degree 3, hence iROZrR is conjugate to z ~ z 3. There are precisely two conjugating maps, since z ~ - z conjugates z ~ z 3 to itself and is the only analytic map other than the identity to do so.

Either conjugating map extends to R0, and this extension has a limit at the co- critical point. The two limits are opposite of each other, and their square is r

Proposition 2.9 shows that the ambiguity is real.

PROPOSITION 2.9. There exists a mapping

r: ~ ' (0-- , ~ ( - ~ ) ,

and for each R ~ ~ r an analytic isomorphism

rR: R--->~(R)

such that / f f=(f~: R~--~ R',)~lv is an isomorphism of patterns, then either R~=R" and f ,=id for all n, or R'=r (R ' ) and fn=rR for all n.

Proof. We construct r and the rR, as usual, by induction. Start the induction by setting r(R0) to be the unique element R6E~0(-~), and defining rRo:R0--.R ~ by rR0(z)=-z. Note that rR0 does conjugate arR0 to :rR~.

Suppose by induction that r and the rR have been defined for all m<n. Choose R E ~n(~), and let S=s~(R), S' =r(S)E ~ - t ( - r IfA is the annulus of S-clos(is(S~_l(S))) defining R, then A' =rs(A) defines an element R' E ~ ( - r The mapping rs extends to a homemorphism rs: ~r162 which can by Lemma 2.1 be chosen to send ramification values to ramification values. Since/~ and R' are standard triple covers of ~r and ~r respectively, the mapping rs lifts to an isomorphism rR: R - , R ' . This constructs r and all the rR.

Note that the construction of r was "unique": given that r(S)=S', there is a unique R' E s,-I(S ') such that there exists an extension of rs: S--~S' to rR:R-->R', and the extension is itself unique.

Now suppose that f=(f,:Rn--*R',),~N is an isomorphism of patterns. Then fo: Ro-*R~ must be either z ~ z or z ~ - z in order to conjugate :rR0 to arR~ since these are the only automorphisms of a neighborhood of oo in C to commute with z ~ z 3.


The result now follows from the uniqueness inherent in the construction of 3, which itself follows from the uniqueness of Lemma 2.2. Q.E.D. for Proposition 2.9

COROLLARY 2.10. I f f'. R| is an isomorphism of patterns o f infinite depth, then either R~=R" and f=id or R'=r(R~) and flR =zR for all n. In that case, we will say that r(R|

Remark 2.11. We can also read ~2 from a pattern R of depth N > I without looking near ~ , as follows. From the proof of Proposition 2.8 we know how to compute r=l~l. Choose a conformal map

~: {zeC}r<lz] < r3}---~ A,

which extends to the boundary such that inner boundary is mapped to inner boundary; such a ~p is uniquely determined up to rotation. There exists two points ~ and ~2 with ~p(~l)=~p(~2)=~, and a unique point to such that ~P(to)=toR. One can check that ~13=~23, so there exists a unique third point ~3 such that ~33=~3=~23. The ratio to/~3 does not depend on the choice of ~p, and is ~2. See Figure 2.6.

~3

A0 to

) t 1

Fig. 2.6

PA'ITERNS 249

2.8. The pattern bundle

The entire construction of ~(r was functorial; ~(~) is defined up to unique isomorphism. As a result we can glue all these structures together to make a "bundle" over C - / ~ .

PROPOSITION 2.12. For any N EZ there exists a unique analytic structure on the disjoint union

UN= U U R

maldng this set into a 2-dimensional complex manifold, such that: (a) The union to=UtoR is for each N>~I a smooth analytic curve; (b) The map :rN: UN--->UN_ 1 induced by the ramified triple covering maps

:rR: R---> s N (R)

is a triple ramified covering space, ramified along the curve to; (c) The complex structure on Uo is inherited from the identification

Uo=

With this structure, the mapping PN: UN--->C-19 which associates ~ to a point x of R E ~N (~) is an analytic submersion.

Proof. One must show that the construction can be done with parameters. This presents no difficulties and is left to the reader. Q.E.D. for Proposition 2.12

We will think ofpN: UN-'->C-D as a bundle of Riemann surfaces where each fiber consists of (1+3N-1)/2 Riemann surfaces. The bundle is not analytically locally trivial: Propositions 2.8 and 2.9 say exactly when two fibers are isomorphic. It is topologically locally trivial, and we will study its monodromy in Chapter 6.

The reader will observe that all the structure of patterns made its way into the construction above, except that no analogue of the mappings iR is mentioned. Recall that the mapping iR: SN(R)--->R is labelled by its range rather than its domain. This is necessary, because for N~>2 a point (~, x) E UN_ l belongs to a pattern R E ~N-I(~) which can be extended to a pattern of depth N in several ways, all of which are of interest. However, there is a mapping defined on a subset of UN_ l, which is a restriction of some JR,, for all the R' with SN(R')=R.

17-928286 Acta Mathematica 169. Impdm(~ le 10 novembre 1992


Fig. 2.7

Each R contains the sub-Riemann surface V(R) consisting of the annulus BN_I(R) and all points inside it, and V'(R)=V(R)-BN_I(R), which is a disjoint union of annuli. See Figure 2.7.

Define for N~>0 the corresponding open subsets V ~ VNC UN of the pattern bundle of depth N by

VN= U U V(R), ~E C-/~ RE ~N(~)

and

V'N= U U V'(R). ~E C-D R E ~N(t;)

By the construction in Section 2.2, if (C,x)EV'(R)~V'N_ ~ there exists a unique R' E~N(~) such that SN(R')=R and iR,(x)EBN_1(R')cV(R'). This defines for N~>I a mapping

iN: V~-l--" VN.

PROPOSITION 2.13. The mapping iN: V~_I---) V N is injective and analytic.

Proof. The map iN is injective since each i R, is injective. The proof of analyticity must go by induction on N since the analytic structure on UN is defined by induction. The space V6 is by definition the set {(r z)l I~l<lzl<lCI3}, and the composition of zq o il is

PATTERNS AND POLYNOMIALS 251

precisely the map (r z)---~(~, z3). Since ~1 is a ramified covering space and it is continuous, this shows that it is analytic.

From the definition of iN and from t'~N(R)O~sN(R)=ZtRoi R we get for each N>~2 that the following diagram commutes

V~N-I

7tN-1 1

V'N_ 2 iN- |

7t N

Since ~N-t and ~N are analytic covering maps the result follows by induction. Q.E.D. for Proposition 2.13

2.9. The quotient pattern bundle

The statements about isomorphisms of patterns in Section 2 have immediate general- izations to the bundles under consideration here.

PROPOSITION 2.14. (a) The mapping rN: UN'--~UN induced by rR:R---~r(R) for each R c UN define for each N an analytic involution covering the involution z~-~-z of C-15.

(b) The group of bundle-automorphisms of Us for N> 1 is {id, ru}.

Proof. The mapping r0: U0---~ U0 is simply (~, z )~( -~ , -z), hence analytic. Now the result follows by induction, as usual. Q.E.D. for Proposition 2.14

In Chapter 6 we will consider the quotient bundle PN: (/N--~C--D, where ON= UN/rZv, and ps([x])=(p~(x)) z so the fiber above ~ in the bundle PN: UN--*C-D is naturally isomorphic to the fiber above ~2 in the quotient bundle PN: Ozv~C-19.

Chapter 3. Patterns and polynomials

Throughout this chapter, P will be a monic cubic polynomial, with critical points tal, tOz satisfying he (~2)<he(o~l). Set


Up (z0) = {z ~ cl hp (z) > hp (z0)}.

In this chapter we will show that subsets Ue(zo) are isomorphic to subsets of patterns when he (o92)<he (z0).

More precisely, suppose he(Wl)/3N+l<~he(w2)<he(w,)/3 N. Then we will show that there exists a r E C - / ) , an abstract Riemann surface R E ~N+,(~) and an extension of ~0e to an analytic isomorphism

~p: U e (w 2) ~ R(P),

where R(P) is an appropriate subset of R. If ~o2 does not escape to o0, we will show that C - K e is isomorphic to an

appropriate ~-pattern of infinite depth. The precise statements are contained inTheorem 3.1 and Corollary 3.5.

If 3.1. Polynomials realizing patterns

he(~ol) he(~ot) 3N+, ~<hp(~O 2) < 3 N

P will be said to have depth N. The polynomial P will have critical depth if the left hand inequality is an equality; if he(w2)=0, i.e. if w2EKe, we say that P has infinite depth. Theorem 3.1 and Remark 3.2 justify this terminology.

THEOREM 3.1. Suppose depth(P)=N~>0. Let ~ be the co-critical inverse image o f P(Wl), and set

lim q~e(z) = ~.

Then there exists a unique ~-pattern R E ~N+l(~) and a unique analytic isomorphism (~e: Ue(w2)--->R(P) where R(P)~-R is defined by

R(P) = {z E RI hR(z) > he(~o2)},

such that the diagram Ue(~o 0 ~ Ue(oJ2)

9 e ee

n o c

commutes, where R0 = (z 6 RI hR(z)>log Ir R(P)

PATTERNS A N D POLYNOMIALS 253

Remark 3.2. Theorem 3. I shows that there is a pattern R of depth N + 1 associated to every polynomial of depth N. We have the following inclusions RIveR(P)= ~v(Ue)cR where the right inclusion is an equality only if the polynomial has critical depth N. Hence we say that the polynomial P is a realization of the ~-pattern R to depth N. The theorem does not assert that every pattern of depth N can be realized by some polynomial of depth N. This is true, and will be shown in Theorem 8.2. To show this, we will need to evaluate q3v at critical values; the Corollary 3.3 below will allow this.

For n<~N we define the critical annulus Cn(P) of P at level n to be C,(P)= c~1(C,(R)), i.e. the connected component of the potential strip of level n surrounding the critical point to2, and we define the critical value annulus B,_I(P) of P at level n - 1 to be B,_I(P)=~,1(B,_1(R)), i.e. the connected component of the potential strip of level n -1 surrounding the critical value P(to2).

Proof o f Theorem 3.1. Set

U n = P - n ( U p ( O ) l ) ) = z E C he(z)>

the proof must go by induction on n since that is the way the patterns are constructed. Set M = N unless P is of critical depth then set M = N + 1. We will construct for

n<.N+ 1 a sequence of patterns R, E ~,(r with s~(R,)=R,_ I and for n<.M a sequence of analytic isomorphisms q0n: U~---~R, such that the diagrams

P U n ~ Un_ 1

(*) ~"[ ~.. [~"-' R n . Rn_ 1

commute. To start the induction set q00=q0v and define r by r 6 cpv(z). If M>0, make the following inductive hypotheses: for all O<m<.M

- - For all n<m there exist unique Rn E ~,(~) and isomorphisms q%: U,---~R, which make the diagrams (*) above commute.

If r/ER satisfies l<r/<3, set U,,~={zEClhe(z)>7?logl~l/3~}; we have the inclusions U._1=U..~cU ..

The restriction q0.,~ of q0n to U.,~ extends as a homeomorphism to q~..,: C--*/~. for all n<m.

We need to introduce r/> 1 for the following reason: each component of C - U . , . is


homeomorphic to a closed disc, and these correspond bijectively to the components of /~n-R~. This is not true if r/= 1; the components of the boundary of U~ are graphs with double points. So in particular, the second inductive hypothesis is not true if r/= 1.

Since m<~N, the point ~m_l,~(P(to2))~l~m_l--gm_l,~l where

Rm_l, ~ = q)m_l,rl(Un, rl)'~

the mapping q3m_l, ~ can be modified so that

x = ~m- l,,/(e(t~ E R m _ 1 - - cloS(Rm- 1)

without modifying it on Un. ~. Then x is separated from oo by a unique component A of Rm_l-ClOs(iRm_t(Rm_2)), which is an annulus nested within Bm_e(Rm_l). This annulus (and the choice of x) selects the element Rm=RA E ~m(~), and by Lemma 2.2 there exists a unique homeomorphism

q,~: C--* R m

such that the diagram

(**)

C

"J '~/+m

P ,,. C

~m-t,+

Rm-I

commutes. If l<rh<r]2 then W~,=~P~2 on Um,~. Therefore q~m=lim~l ~ is well defined and makes the diagram (*) commute. Moreover, the maps ~,~ are extensions verifying the second inductive hypothesis.

If P is of critical depth set R=RN+ 1 and ~e=q~N+l. If P is not of critical depth then choose r/= 3 N+ l h e (tOE)/h P ((/)1) SO that UN+ 1, ~ = UP (t~ Construct RN+ 1 and ~p~: C-->/~N+ 1 as above and set R=RN+ 1 and ee=W, lu~) . The sets Ro=slo...OSN+l(R) and {zERI hR(z)>log I~1} are canonically isomorphic so we can identify them.

If there were another pattern R' satisfying the requirements of the theorem, then the isomorphism q3J, would induce isomorphisms q0" at all levels. By hypothesis, cp~=Cpo=q~e, and now the equality q0m=q0m follows by induction from the uniqueness in Lemma 2.2. Q.E.D. for Theorem 3.1

PATTERNS AND POLYNOMIALS 255

COROLLARY 3.3. Suppose depth(P)=N>~O. Then the critical value P(to2) lies in Up (a~2) hence is in the domain o f (De.

Remark 3.4. Theorem 3. I and Corollary 3.3 can be extended to the case where he(tol)=he(tO2) and OJl=~W2 (i.e. critical depth -1) . The mapping qge can then be extended to a neighborhood of the co-critical point oJi and it is still true that ~ E C - / ) is uniquely determined. However, ~ depends not just on P but also on the order of the critical points. Hence Theorem 3.1 and Corollary 3.3 do not apply to the case where to1=co2, there is then no co-critical point, so ~ is not defined.

COROLLARY 3.5. Suppose depth(P)=oo. Let wi be the co-critical inverse image of P(wl), and set ~=limz_,o,,gp(Z). Then there exists a unique ~-pattern R| of infinite depth and a unique analytic isomorphism ~e: C-Ke--->R~, which extends ~Oe. The isomorphism ~p conjugates P to :rR|

The proof is essentially identical with the proof of Theorem 3.1.

Remark 3.6. Corollary 3.5 shows that there is a pattern R= of infinite depth associated to every polynomial of infinite depth. We say that the polynomial P is a realization of the ~-pattern R|

The corollary does not assert that every pattern of infinite depth can be realized by some polynomial of infinite depth, but this is also true, and will be shown in Theorem 9.1.

3.2. Introducing parameters in Theorem 3.1

Let A be a complex manifold, and let (P~)aeA be an analytic family of monic cubic polynomials parametrized by A, and with distinct critical points co1(2), co2(A) satisfying

h~(~o1(2)) h2(r < 3 N '

i.e. all Pz have depth ~>N. Let w~(2) be the co-critical inverse image of Pa(wl(2)). Set

~(A) = limz_~ .(~ ) q~(z),

and


Define c~x=r ~ (from Theorem 3.1), and similarly q%,~ the mapping q~n constructed in the proof of Theorem 3.1, as applied to the polynomial P~.

The spaces tin defined in Section 2.8 have the following universal property:

PROPOSITION 3.7. The mapping r I given by

[q3a(z) if depth(P~) = N r Z) /

(q0N+l,a(Z) if depth(Pa)>N

is analytic. Let p: XN+I---~A be the projection onto the first factor. Then the diagram

XN+I r ~ UN+ 1

commutes. I f depth(Pa)<~N+ 1 then the critical value Pa(to2(2)) E XN+ 1 and q~(Pa(wz(2))) E VN+ r

Proof. Define ~:X~---~U, by cp~(2,z)=qg~,~(z). The proof goes by induction on n=O ..... N+ 1, essentially putting parameters in the proof of Theorem 3. I. To start the induction, observe that ~0(~., z)=~pa(z), which depends analytically on (2, z).

The diagrams

X. Ca, z) ~ (~.P~(z)) ~, X . I

Ui 1 I~ U~I_ 1

commute, since they commute on fibers. Since the diagram commutes and since q~: X~-~ U~ is a lifting of tpn_ 1 to a ramified

covering space, it follows by induction that all tp~'s are analytic for n<.N+ 1. Q.E.D. for Proposition 3.7

ENDS OF PAITERNS 257

Chapter 4. Ends of patterns

In Section 2.2 we defined the tree of Riemann surfaces ~(r and in Section 2.6 the patterns in ~ ( ~ ) of infinite depth. In this chapter we will investigate the ends of the Riemann surfaces of infinite depth.

In Section 3.1 we defined cubic polynomials of infinite depth and their associated patterns. In Chapter 5 we shall consider cubic polynomials of infinite depth. The components of the filled-in Julia set Ke of such a polynomial P are in 1-1 correspondence with the ends of the pattern R= associated with P. We will discover in Chapter 5 exactly when a component in Ke is a point and when it is a continuum. The two cases correspond exactly to the two types of ends analyzed in this chapter: divergent ends and convergent ends respectively.

4.1. Patterns and their ends

In this section we will only be concerned with patterns of infinite depth, and will omit the words 'infinite depth' throughout. Recall that to each such pattern we can associate the Riemann surface

R= = ~ (R,,, iR.+, )

and in this section we will drop the subscript oo. The Propositions 2.8 and 2.9 say that R contains all the information about the pattern up to determination of +r most often we will identify a k-pattern and the corresponding Riemann surface.

We will be mainly concerned with the set of ends E(R). Of course oo is an end of R, but we don't want to consider it, so we set

e ( R ) =

where/~=R tJ {oo), and the projective limit is taken as usual over the directed system of compact subsets K of/~.

The topological set E(R) is a Cantor set, and since the mapping zrs: R-.-,R is proper, it induces a continuous mapping E(:rR): E(R)-->E(R), which we will ordinarily still denote zrR.

The subsets clos(/~,) form a compact exhaustion of/~, so this space of ends can also be described as

E(R) = ~ ~o(R-Rn).


4.2. Nests and tableaux

Let R be a ~-pattern. For any x E E(R), define the nest N(x) of x to be the sequence A I (x)0~<t<| of annuli of R, where At (x) is the annulus at level I separating x from ~ . The nests of any two ends of R coincide up to some positive level and then differ from there o n .

Set

mod N(x) = ~ mod (A t (x)). /=0

We will call the end x E E(R) convergent if

mod N(x) < divergent otherwise. If we realize R as an open subset of C, with oo at oo, through a polynomial P, then each end of R will correspond to a component of C - R , and conversely. We will see in Section 5.3 that if an end is divergent, then the corresponding component must be reduced to a point. As we will see in Theorem 5.2 it is also true for a cubic polynomial P that if an end is convergent then the corresponding component is not reduced to a point.

One end of any ~-pattern R is distinguished: the critical end c E E(R), whose nest, the critical nest, N(c) consists of the critical annuli Co, CI . . . . . which map to their images under :rR by double covers.

In order to estimate modN(x) for an end xEE(R), we will define its tableau T(x); this is the two dimensional array of annuli At. k(x), where At, 0(x)=At (x) is the annulus of N(x) at level l, and Al, k(X)----~k(Al+k,o(X)). Another way of saying this is that the kth column is the nest N(yt~k(x)) of :r~k(X) SO that Al, k(X)=Ai, o(Yt~k(x)). See Figure 4.1.

N(x) N(:tR(x)) N(:t~k(x))

Ao, o(X) AI,o(X)

At, o(X)

Ao, l(x) , r

,,, At, 1(x)

, At, l(x)

f

f ~ 1 7 6 1 7 6

f i

Ao, k(x) Al k(x)

At, k(x)

f

f

f

f

R r

. ~

f

Fig. 4.1. The tableau T(x).

ENDS OF PATTERNS 2 5 9

Since Co is the only annulus at level 0 we have Ao.k(x)=C o for all k and all x. The map ~R: At, k(X)~At-l, k+l(x) is a double cover if Ai, k(X)= C I and a simple cover

otherwise. The critical tableau T(c) is the tableau of the critical end. Of primary interest in a tableau are the critical positions, i.e. the pairs (l, k) such

that At, k(x)= C r For instance, the critical end is periodic of period dividing k if and only if the kth column of the critical tableau is entirely critical, and an end x is a preimage of the critical end if and only if some column of T(x) is entirely critical.

All moduli of annuli can be computed from the critical positions. The modulus of Ai, k(X) satisfies

k(x) ) = 1 mod(C0). mod(A/,

where n is the number of critical positions (i,j) along the diagonal i+j=l+k and with O<i<.l.

4.3. Properties of tableaux

PROPOSITION 4.1. All the tableaux for a given E-pattern satisfy the following 3 tableau rules:

(Ta) The critical positions form unbroken vertical lines stretching out from the top of a tableau, i.e. i f Al, k(X)=C l then Aj, k(X)=C J for all O<~j<~l.

(Tb) I f At, k(X) = Ct then Ai, k+j (x)--Ai, j (C) for i+j<~i; (Tc) I f for the critical tableau T(c) and for some l>~n>O

Al+l_n,n(c)=Ct+l_ n and Al_i,i(C)::~=Cl_ i for 0<i<n, and if for any tableau T(x) and for some m>0

Al, m(X)=C t and At+l,m(X)*Cl+l, then Al+l_n, m+n(x)4=Cl+l_n.

Proof. Properties (Ta) and (Tb) are fairly clear: (a) The vertical columns are nests, which always agree up to some point with the

critical nest and then separate. (b) Down to level I the kth column in the tableau T(x) is the same as the 0-column of

the critical tableau. A complete triangle is copied from the critical tableau to the tableau T(x), since

Ai, k+j (X) "-~ Yr~J(Ai+j, k(X)) -~ y[~J(ci+j) ~-- y[~J(ai+j, 0(c)) = ai, j (c).

Property (Tc) is more technical; it is seen as follows:

260 B. BRANNER A N D J. H. HUBBARD

* o 9 v

,(!,0). . T(c)

l-- ,o 9

:Z:: ,(l, k ) .

9 T(x)

Fig. 4.2. R u l e (Tb).

(c) The first part of the hypothesis says that the image of Ct under ~" is Ct_ ., that ~ " restricted to Ct and all points inside it is of degree 2 and that the inverse image of Ct+l_ . under this restriction has Ct+ ~ as one component. Now since the degree of the restriction is 2, the Ct+ ~ is the entire inverse image, so there cannot be in addition a non- critical annulus in the inverse image. Q.E.D. for Proposition 4.1

Most often when discussing a tableau for a given ~-pattern we will disregard all information except the critical positions, and think of it as simply the grid N 2 with critical positions marked. The rules (Ta), (Tb), (Tc) above can be formulated in terms of marked grids. Of particular interest is the critical marked grid with all positions in the 0-column marked. We will mark a critical position by 9 and a non-critical position by o. The tableau rules (Tb) and (Tc) are illustrated in Figures 4.2 and 4.3 respectively.

t ' - o o o

o o o

o o o o

o o o o / , 0 0 0 0 ,/, o o o

/ o o

7' o

(l+ 1,0)

~ ~

o o

o

0 0

0

(1+ l - n , n)

T(c)

o

o

o

9 o o

o o

9 o o

o o /

o

{,m) 9 o ( l + l , m )

o o q

o o I

o o

o o q / , 0 .0

/ o /

ii: 0

(l-n, re+n)

o (l+l-n, re+n)

T(x)

Fig. 4.3. Rule (Tc).

ENDS OF PATrERNS 261

4.4. The realizability of tableaux

Properties (Ta), (Tb), (Tc) of tableaux capture much of what the dynamics actually allows, as the following proposition shows.

THEOREM 4.1. (a)Any critical marked grid G satisfying the rules (Ta), (Tb), (Tc) can be realized for any ~ E C-19 as the critical tableau for some pattern R E ~| in general the ~-pattern is not uniquely determined from the marked grid.

(b) Given a pattern R E ~ ( ~ ) , any marked grid satisfying (Ta), (Tb), (Tc) with respect to the critical tableau o f R can be realized by an end o f R. In general the end is not uniquely determined from the marked grid.

Proof. (a) The proof goes by induction. A pattern Rn E ~n(~) has a restricted critical tableau (Aid (c))i+j<~n; suppose we have found one whose critical positions (i,j) coincide with our critical marked grid G for i+j~n. We will show that there is at least one choice of Rn+ 1E ~+1(r with sn+l(R~+l)=R~ whose restricted critical tableau has critical positions agreeing with those of G in the range i+j<~n+l. We shall show this by choosing annuli Ai,~+l_i(c ) of the pattern R~ for i= 1 .. . . . n; the choice of An, i(c) will tell us which annulus the critical value annulus at level n should be and hence determine R~+ v

First, let (k, n + l - k ) be the critical position on the line i+j=n+l with the largest k, O<.k<~n; in the position (i, n+ 1 - 0 with O<.i<.k recopy the annuli in positions (i, k - 0 . By property (Tb) these positions will be critical if and only if the corresponding positions of G are critical, moreover the choices are compatible with :tRo and nesting.

Now successively fill the positions ( j + l , n - j ) for j=k , k+ l ..... n -1 . This can be done without ambiguity so long as the position (j, n - j ) is not critical. In that case :tRn: A j, ~_j (c)--~Aj_ 1, ~-~+1(c) is of degree 1 so there is a unique annulus nested in Aj, ~_~(c) and mapped onto Aj,~_j+l(C) by :rRn. If the position (j, n - j ) is critical, the analysis is involved; we will call such cases ambiguous cases.

Suppose ( j + l , n - j ) is an ambiguous case and let m>0 be that integer such that ( j - m , m) is critical and all the positions ( j - i , i) for 0<i<m are non-critical, see Figure 4.4. By property (Tb) the position ( j - m , n - j + m ) is critical. The analysis divides into 2 cases according to whether the position ( j - m + I, m) is critical or not.

If the position ( j - m + l , m ) is critical then by property (Tc) the position ( j - m + I, n - j+m) is not critical. The annulus Aj, n_j+l(c) is different from Aj, 1(c), since ~m-l)(Aj, l(C))=Cj_m a n d ~m-l)(Aj, n_j+l(c))=l:Cj_m, so it has two (non-critical)inverse images in Ay, n_j(c)=C i. Hence the annulus Ai+l,~_~ (c) can be chosen in two ways, but as we shall see we may have to change our choice at the next ambiguous case.


(j, o)

(n, 0) (n + 1,0)

/- O r

o 7

0

/ • (j, I)

f f

,1I (j-m, m) o ~<(j-m+l,m)

/ /

/ /

/ /

/ /

/

/ /

/ /

/

f

o f

0

/ ~ -m, n-j+ m) 7 o ~((j-m+l,n-j+m) t ' o / ,

r215 (j, n-j+ I) /

: o ( j + I , n - j )

/

G / /

Fig. 4.4

If the position (j-m+l,m) is not critical then there are 2 cases according to whether ( j - m + l , n-j+rn) is critical or not.

If (j-m+ I, n-j+m) is critical and (j-m+ 1, rn) is not, then the annulus Aj, n_j+l(c) is different from As, i(c), since

J~m-D(Aj , I(C)) . Cj_ m a n d ~ m - l ) ( A j , n_j+l(C)) = Cj_ m,

so it has two (non-critical) inverse images in Aj,~_j(c)=Cj. As above the annulus Aj+l,n_j(c) can he chosen in two ways.

If neither (j-m+ 1, n-j+m) nor (j-m+ 1, m) is critical, then change if necessary the choice of the annulus Aj_m+l,n_j+m(C ) to make it different from Aj_m+l,m(C). This will further change the Aj_m+l+i,n_j+m_i(C ) for O<i<m, in an unambiguous way. Then the annulus Ai,~_y+i(c ) is different from Ai, l(c), since

ENDS OF PATrERNS

!:ii" o

o o

O O O

0 0

0

o . .

263

Restricted critical marked grid; i+j<~ 6.

Fig. 4.5

~ m - ' ( A j , ,(c)) = Aj_m. , . . ( c ) * Aj-m+,,n-j+m(C) = ~$~-'(Aj, n_j.l(c)). Hence as above the annulus Aj+L,_j(c) can be chosen in two ways. Moreover, we will never have to change Aj_,,,+l,,_j+m(C ) again.

(b) The proof is almost identical with the proof of (a). By induction on n we choose annuii {Ai,j}i+j~ ~ of Rr Suppose this has been done for the positions {(i,J)}i+~. Let

One realization of the restricted critical grid in Figure 4.5.

Fig. 4.6


(k, n+ l - k ) be the critical position on the line i+j=n+ 1 with the largest k, O~k<~n+ 1. Then we have to choose Ai.n+~_j=Ai, k_i(c), O<.i<~k. If k < n + l we fill the positions ( j+ l , n - j ) successively f o r j = k .. . . . n. This can be done without ambiguity so long as the position (j, n - j ) is not critical. If ( j+ 1, n - j ) is an ambiguous case, i.e. if (j, n - j ) is critical, let m>0 be that integer such that Aj_m. re(c)= Cj_m and Aj_i,i(c):~Cj_~ for 0<i<m. Similar to the proof of (a) we can always arrange that Aj, n_j+~:Aj, 1(c), but we may have to change the previous choice Aj_m+t.n_j§ m. Q.E.D. for Theorem 4.2

Example. Figure 4.5 shows a restricted critical grid with marked positions for i+j<~6 and Figure 4.6 shows a choice of annuli for i+j<~5. For each ~ the grid can be realized by two C-patterns in ~s(~), since the annulus A2.1(c) can be chosen in two ways. I fR E ~5(~) is as chosen in Figure 4.6 and if we choose A2,4(c)=A2, l(C) we are in a situation where we have to change that choice. Therefore we are forced to choose A~,4(c) as shown and there is a unique extension from R E ~5(~) to a E-pattern in ~6(~).

4.5. Moduli of nests

We shall now estimate mod N(x) for an end xEE(R) rising its marked grid. Theorem 4.3 below gives a complete answer to the problem. The proof appears after Lemmas 4.4-4.6, which it requires.

THEOREM 4.3. (a) The critical end cEE(R) is divergent if and only if it is not periodic.

(b) I f the critical end c E E(R) is divergent then any end x E E(R) is divergent. (c) I f the critical end c E E(R) is convergent then an end x E E(R) is convergent if

and only if it is precritical, i,e. i f there exists an n with zt~n(x)=c.

LEMMA 4.4. I f xEE(R) is a convergent end, then to the right o f any position in its tableau there is a critical position.

Proof. The modulus of At. 0(x) is expressed by

mod(At ' 0(x) ) = 1 mod(C0), 2 n

where n is the number of critical positions along the diagonal i+j=l with O<i<.l. If the horizontal line beginning at the position (m, k) has no critical positions to the right of (m, k), then by rule (Ta) the position (i,j) is non-critical whenever i>-m andj>k. Hence n<.m+k. For any l

ENDS OF PATrERNS 265

mod(At, o(X)) I> ~ m~

and clearly the end is divergent. Q.E.D. for Lemma 4.4

We will call the critical end recurrent if its tableau satisfies the conclusion of Lemma 4.5, i.e. if to the right of every position there is a critical position.

Define a position (l, k)=#(0, 0) of the critical tableau to be an originator if and only if either

l--0 and there does not exist an i, 0< i<k with (k- i , i) critical o r

l>0 and (i) (l, k) is a critical position; (ii) there exists i with 0< i<k such that (l, i) is critical; (iii) all (l+i, k - i ) for 0< i<k are non-critical.

Clearly (0, 1) is an originator. We shall mark an originator by |

LEMMA 4.5. I f the critical end cEE(R) is recurrent, then

mod(N(c)) = ~ mod(C) = rood(C0)+ E m~ k (c))" j r0 (I, k) originator

Proof. For each originator consider the subsequence of the critical nest constructed as follows: go down diagonally to the 0th column, then horizontally until you reach a critical annulus which exists since c is recurrent, then down diagonally until you reach the 0th column, etc. (see Figure 4.7). The moduli of the annuli on the 0th column you meet this way form a geometric series with ratio 1/2, which starts with 1/2 the modulus of the originator, thus the series sums to the modulus of the originator. Clearly each annulus except Co of the critical nest is reached exactly once in this way.

Q.E.D. for Lemma 4.5

The next result is really the crucial point.

LEblblA 4.6. I f the critical end c E E(R) is recurrent and non-periodic, then there is an originating position in every row.

Proof. By induction, suppose that the annulus at the position (l, k) is critical and originating, and let (l+ 1, k') be the first critical annulus on or to the right of (l+ 1, k). If in the l+ 1 row in position 1 . . . . . k -1 there are any critical annuli, then ( l+l , k') is

18-928286 Acta Mathematica 169. Imprim~ le 10 novembre 1992


f "7 9 " /

. ," /

. , ' " / / / , ," / '

/ ." /

.." /"

/ /

/

. . . "

Fig. 4.7. The critical tableau.

originating. So assume there are none. The kth column is not entirely critical, since the critical end is assumed non-periodic, so continue the line of critical positions through (l+ l, k') to its end (l', k') and consider the diagonal just beyond its tip, i.e. the line of positions (i,j) satisfying i+j=l'+k'+ I, see Figure 4.8.

Claim. There are no critical positions on this line with first coordinate greater than l+l .

If there were such a critical annulus, then by property (Tb) of tableaux it would have to appear in some position

(l'-(m-l)k'+l,mk'), for some l<m<(i'-l+k')/k';

let m0 be the smallest m for which this occurs. Then w the positions ( l ' - (m0-1) k', k'), ( l ' - ( m o - I) k' + 1, k') are critical and none of the

positions (l'-(mo-2)k'-i, 0 are critical for 0<i<k' , the position ( l ' - (m0-2) k', (m0-1) k') is critical and the position

( l ' - (m0-2) k'+ 1, (m0-1) k') is not critical. Thus position (l'-(mo-1)k'+ 1, mok') is also non-critical by property (Tc). This

contradicts the hypothesis about m0 and prooes the claim.

0 0 0 0 0 0 0 0

0

0

0

0

(l+ 1 +k', O)

0

0

0

o

(l', k'l O

ENDS OF PATI'ERNS

(l+ 1, k) 0

0

0

0

0

0

0

267

Fig. 4.8

Now we see that the first critical position on or to the right of the position (l+l,k'+l'-l) , which exists since the critical end is convergent, is an originating position. Q.E.D. for Lemma 4.6

Proof of Theorem 4.3. (a) If c is periodic, it is clearly recurrent, and has only finitely many originators. So it is convergent by Lemma 4.4.

If c is recurrent and non-periodic, define the function t: N*---~N by the rule that (t(n),n-t(n)) is critical and (m,n-m) is not critical for t(n)<m<n, so that mod(Ct(n))=2 mod(Cn). Lemma 4.6 says exactly that for any level n, there are at least two distinct levels n~ and n2 such that t(nO=t(n2)=n, so that

mod(C.) ~< E mod(Cj). t ( j ) f n

The function t is strictly decreasing, so we can define the generation of n to be that k such that f (n)=0. It follows that

mod(Cj)~ < ~ mod(Cj), gen(j)ffik gen(j)=k + I

and Theorem 4.3 (a) follows, since there are infinitely many generations.


(b) Suppose the end x E E(R) is convergent. For any l>0, let (l, k(l)) be the first critical position in row l of the tableau T(x) with k(l)>.O. By Lemma 4.4 this position exists. The critical positions (l, k(l)) for l>0 form the critical staircase of the tableau of x. Then

mod(At+k(t),0(x)) = mod(Ct).

Moreover, the integers l+k(l) are all distinct. Thus

mod N( c) <~ mod N(x) < oo,

which is a contradiction. (c) If x E E(R) is precritical it is clearly convergent. Suppose x E E(R) is convergent and consider the critical tableau. By (a) it repeats

with some period k. There are no critical positions in the tableau beneath row l for some 1, except in the columns with column-number divisible by k. Set

k+l

S = E mod(Ci). i=1+1

The critical staircase of the tableau of x is the set of critical positions in T(x) which have no critical positions to the left in the same row.

If :t~P(x)~=c for all p and modN(x)<oo, then the staircase of the tableau of x has infinitely many steps. We will show that every step whose tip has vertical coordinate larger than l+k contributes at least S to the modN(x); there clearly must be infinitely many of these.

The argument is similar to the one in the proof of Lemma 4.6. If (m, n) is the tip of a step with m>l+k, then the diagonal i+j=m+n+ 1 can contain no critical position for j>l. Consider the first critical positions to the right of each of

(n+m- l , 1+ 1) . . . . . ( n + m - l - k + 1, l+k).

The sum of the moduli of these positions is S. The diagonal drawn from each of these positions to the 0-column ends on an annulus with the same modulus as the critical annulus from which the diagonal left. Since all these endpoints are distinct, it follows that x is divergent and we are done. Q.E.D. for Theorem 4.3

JULIA SETS OF CUBICS

Fig. 5.1

269

Chapter 5. Julia sets for cubic polynomials

In this chapter we will completely settle the question of when the Julia set of a cubic polynomial is a Cantor set. P. Fatou and G. Julia proved the following result.

THEOREM (Fatou [F], Julia [J]). (a) The Julia set for a polynomial P is connected if and only if none of the critical points escape to infinity under iteration.

(b) The Julia set for a polynomial P is a Cantor set if all the critical points do escape to infinity under iteration.

Figures 5.1 and 5.2 show a Julia set for a cubic polynomial P satisfying (a) respectively (b) of the theorem.

Fatou conjectured [F, p. 84] that condition (b) above is necessary for the Julia set to be a Cantor set, but this was disproved by H. Brolin [Br], who showed that if P is a real cubic polynomial with one critical point ~Ol escaping to ~ and the other critical point tOE being mapped to a fixed point not in the component of Ke which contains to2, then Ke is a Cantor set. Figure 5.3 shows the graph of such a cubic polynomial. The Julia set is a Cantor set on the real axis.

Using tableaux, it is easy to reprove Brolin's result, and more generally to show that when one critical point ~Ol escapes, the other to2 does not and the component of Ke containing to2 is strictly preperiodic, then the Julia set is always a Cantor set. As we will see this follows from Theorem 4.3 (b) and Lemma 4.4 since in that case the critical end is not recurrent.

Tableaux and originators were invented in order to prove Theorem 5.2 below in the


Fig. 5.2

cases where the critical component is recurrent. As we will see in Chapter 7, this result has consequences in the parameter space also.

In order to show that some Julia sets are not Cantor sets, we will require polynomial-like mappings [DH2]. Recall that a polynomial-like mapping of degree d is a triple (U, U',f) where U and U' are plane domains homeomorphic to discs, with U' relatively compact in U, and f : U'---~ U is analytic of degree d. The filled-in Julia set Ky of the polynomial-like mapping f is defined as

Kf= (zE U'If~ U' for all n}.

. ~

t O 1 0)2

Fig. 5.3

Fig. 5.4

JULIA SETS OF CUBICS 271

We say that a point z in U' does not escape under iteration by fexac t ly when z is in Kf. As we will see, there are many iterates of cubic polynomials which are polynomial-

like of degree 2; one example stands out particularly. If P is a cubic polynomial with critical points wl, w2 satisfying he (to2)<he (oJ 0, then the locus V= {zl he (z)<he (toO} has two connected components, one of which contains to2. Call this one U' and the other U". Let U=P(U')={zl he(z)<3he(toO}. Then f=PIv': U'--->U is polynomial-like of degree 2. See Figure 5.4.

Remark 5.1. It is important to consider the domain of definition U' of a polynomial-like mapping f as part of the definition. For instance, the statement that the critical point of the polynomial-like mapping does not escape is much more restrictive than saying that this critical point has a bounded orbit under P; for example P(to2) might be the fixed point of P which lies in U"; this is precisely what happens in Brolin's example.

We shall need the definition of hybrid equivalence and the straightening theorem from [DH2].

Two polynomial-like mappings (U, U',f) and (V, V',g) of degree d are said to be hybrid equivalent if there exists a quasiconformal homeomorphism tp from a neighborhood of Kf onto a neighborhood of Kg, conjugating f and g and so that aq~=0 on K s.

THE STRAIGHTENING THEOREM ([DH2]). (a) Every polynomial-like mapping (U, U ' , f ) of degree d is hybrid equivalent to a polynomial of degree d.


(b) l f Kf is connected, then the polynomial is uniquely determined up to conjugation by an affine map.

5.1. The main statements

Let P be a monic cubic polynomial with one critical point (D~ escaping to o0 and the other critical point (D E not and with limz__,,o, ~ q~e (z)=~ 9 Let R| be the associated E-pattern and $e: C-Ke--~R| be the analytic isomorphism extending q0e, as constructed in Corollary 3.5.

The components of the filled-in Julia set Ke for the polynomial P are in 1-1 correspondence with the ends of R| In fact another way of understanding the set E(C-Ke) of ends is as follows: consider the equivalence relation on Ke:

x ~ y if and only if x and y are in the same component of K e.

The quotient space Ke/~ with the quotient topology is homeomorphic to E((2-Ke); hence it is a Cantor set.

As we shall see, if the critical component of Ke is periodic then the set Ke has countably many components homeomorphic to Ka for some a E M and uncountably many point components. (Recall that M={a I the Julia set of z-->z2+a is connected}.)

We shall denote the component of Ke corresponding to x E E(R| by Ke (x). For any x E E(R| we have P(Ke (x))=Ke(3rR (x)), i.e. the restriction of P to such a component maps onto the image component. This is a consequence of P being proper, with everywhere positive local degree.

We call the component Kp (x) periodic, preperiodic or strictly preperiodic if the end x has the corresponding property. If x is periodic of period k, the annulus An(x) E N(x) at level n is mapped to An-k(x). Thus each periodic end, in addition to its period k, has a level n(x) which is the smallest integer n such that An_j(x)g=P~ for j=l ,2 ..... k-1 .

The component Ke(c) corresponding to the critical end c is called the critical component; of course (D2 EKe (c). Let Wn be the smallest simply-connected domain in C containing the critical annulus Cn(P). Then each Wn is homeomorphic to a disc and Wn is relatively compact in Wm for each m<n.

Theorem 5.2 describes the Julia set if the critical component is not periodic, and Theorem 5.3 describes the Julia set when it is periodic.

Let P be a cubic polynomial with one critical point ta~ escaping to oo and the other (D2 not.


THEOREM 5.2. The Julia set Jp is a Cantor set if and only if the critical component is not periodic.

THEOREM 5.3. Suppose the critical component is periodic o f period k and level no. (a) There exists a unique quadratic polynomial o f the form Q(z)=z2+a with a E M,

such that the polynomial-like mapping

f ~ ok . P I%. W o-k

is hybrid equivalent to Q. (b) A component o f Ke is quasi-conformally homeomorphic to KQ if it is a

preimage of the critical component o f Ke and is a point otherwise.

5.2. Analytic preliminaries In Theorem 5.2 and Theorem 5.3 it is stated precisely which components of Kv are points. In order to prove that we shall need the following two classical results [A]; the proofs in the litterature depend on extremal length, and we will give a different one.

PROPOSITION 5.4 (Gr6tzsch). Let (Aj) be a (finite or infinite) sequence o f open annuli. Let qgj: Aj--->A be conformal mappings which are homotopy equivalences, with disjoint images. Then

X mod(A) <~ mod A.

Fig. 5.5

h


Proof. Let B(h)= (zl 0<Im(z)<h}, and write

A = B(h)/Z, Aj-- B(hj)/Z

so that h and hj are the moduli of the corresponding annuli. Parametrize the annuli by z=x +iy, zy=xj+iyy. Then

h=fadxdy~X~j %~A)dXdy=~j failq~J'(zJ)12dxjdyj=X fohJ(follq~J'(zJ)12dxj) j

>~fohJ(fo1[CPj'(z)ldxj)2dyj>~fohJdyj~Xhj 9

The first inequality follows from the fact that the Aj are disjoint, the second is Schwarz's inequality, and the third is that the image of all "equators" of annuli Aj have length greater than 1, since the inclusions q0 i are homotopy equivalences.


PROPOSITION 5.5. Let AcD be an open annulus of infinite modulus. Then the bounded component of C-A is a point.

Proof. If the component contained two points x, y, then the geodesic in D- (x, y} (for its hyperbolic metric) in the homotopy class of A would be shorter than any curve in A in that homotopy class. However, there are arbitrarily short curves in A in that homotopy class. Q.E.D. for Proposition 5.5

5.3. Proofs of the theorems

Proof of Theorem 5.2. Using Theorem 4.3 (a) we have that the following three statements are equivalent: P~ is not in Ke(c) for any n>0, the critical end cEE(R) is non-periodic, the critical end is divergent. If this is satisfied it follows from Theorem 4.3 (b) and Propositions 5.4 and 5.5 that all components of Ke are just points. This proves that non-periodicity of the critical end implies that Kp is a Cantor set. Hence the Julia set Je=Ke is a Cantor set. The converse is contained in Theorem 5.3, proved below. Q.E.D. for Theorem 5.2, ~=

Remark 5.6. If the critical component is strictly preperiodic, then it is non- recurrent. Thus we do not need the full power of Theorem 4.3 (a) for the proof: the critical nest is divergent by Lemma 4.4, and all others also by Theorem 4.3 (b). Hence


in that case the Julia set is a Cantor set, reproving Brolin's result in greater generality. Notice that the proofs of Theorem 4.3 (b) and Lemma 4.4 do not require the third condition on tableaux.

In the case where the critical component is periodic we know that the critical end is convergent. However, Gr6tzsch' inequality in Proposition 5.4 goes in the wrong direc- tion, and would still allow Ke (c) to be a point. That this is in fact not the case follows from the theory of polynomial-like mappings.

PROPOSITION 5.7. (a) Suppose there exist n and k with k<-n such that P satisfies depth(P)~>n, that P~ ) and that P~ for 0<j<k. Then the restriction

f . = P~ W.--* Wn_ k

is polynomial-like of degree 2. (b) I f the critical end is periodic of period k and level no, then the hypotheses above

hold for all n~no and the critical point o92 does not escape for the polynomial-like mapping f no.

Proof. Clearly f~ maps the boundary OWn to the boundary OW~-k. The mapping P: Wn--~P(Wn) is of degree 2, and all the other restrictions of P which appear infn are of degree 1. This verifies (a).

For (b), the same argument as above shows that fn is polynomial-like for all n>-no; this implies that the image fn0(~o2) lies in N, U~=K(fn). Q.E.D. for Proposition 5.7

Fig. 5.6

276

o 9

B. BRANNER AND J. H. HUBBARD

"O~ . ~ 9 . b . 9 ~

9 r

o O"

9 i .,o;. ( ' O I . P(tol)

-o. Fig. 5.7

Example 5.8. Suppose a polynomial P has depth >I , and that the critical value lies in the small lobe U" (cf. Figure 5.4). Set U" to be the component ofP-~(U ") contained in U'. Then po2: U"--~U is polynomial-like of degree 2. (See Figure 5.6.)

Proof of Theorem 5.3. Part (a) follows immediately from Proposition 5.7 and the straightening theorem.

Now consider a convergent end x E E(R). Let n be the smallest integer such that ~r~n(x)=c; the existence of such an n is guaranteed by Theorem 4.3 (b). Then pon maps Ke(x) onto Ke(c) homeomorphically. A component Ke(x) for which the end xEE(R) is divergent is of course just a point (Propositions 5.4 and 5.5). Q.E.D. for Theorem 5.3

Q.E.D. for Theorem 5.2

Examples of Julia sets Jr for some polynomials P with periodic critical component are shown in Figures 5.7-5.12. In Figures 5.7, 5.8, 5.9 the polynomial P satisfies P(to2)=to2 and arg(~e(tol)) equals 0, 1/3, 2/3 respectively. The unique quadratic polynomial associated to P due to Theorem 5.5 (a) is Q(z)=z 2.

In Figures 5.10, 5.11 the polynomial P sa t i s f ies P~ P(092)#092,

arg(q3e(tol))=0 and P(to2) belongs to U', U" respectively (with the notation used in Figures 5.3 and 5.4). The critical component of the polynomial P in Figure 5. I0 has period I and the unique quadratic polynomial associated to P is Q(z)=z 2-1. The critical


L ? ; % r

%~ 2 - - - - P(~~ ~ ~ ~ ~ -" / ~ ~--, o-.r

Fig. 5.8 Fig. 5.9

component of the polynomial P in Figure 5.11 has period 2 and the unique quadratic polynomial associated to P is Q(z)=z 2.

In Figure 5.12 the polynomial P satisfies P~ is fixed, P(o92)~P~ arg(q~p(tO2'))=0 and the period of the critical component is 1. The unique quadratic

9 " o " " " " " .

- ~o, " i , ( ~ o , )

Fig. 5.10


I

(/)1 i \ ! ,

9 , ~ , to I P(~ ! |

0.) 2 = P2((D2)

Fig. 5.11

P(o~ 1)

polynomial associated to P is Q(z)=z2-2. The polynomials used in Figures 5.7, 5.10, 5.11 and 5.12 are all real.

5 . 4 . T h e m e a s u r e o f J u l i a se t s

In this section, we will show that those Julia sets which are Cantor sets by Theorem 5.2 are of measure zero. The proof is due to Curt McMullen, who kindly agreed to let us include it in our paper.

THEOREM 5.9. l f P is a polynomial with non-periodic critical component then the Julia set is a Cantor set o f measure O.

First, a preliminary lemma, very similar to Proposition 5.4.

PROPOSITION 5.10. Let D e C be a simply connected open set, K c D a connected compact subset and A the annulus D - K . Then

4:tmod(A) ~< area(D____)) 1. area(K)

p2(oJ2 ) = P3(~o2) e ( ~ 2 ) .~

Fig. 5.12


Proof. If K has area zero, the proposition is trivially true. Suppose area(K)*0 and scale so that area(K)= 1/4:r, which is the area of the disc bounded by the circle of circumference 1.

Let B(h)={zlO<Im(z)<h}, and B=B(h)/Z so that if h=mod(A), there exists a conformal isomorphism q~:B---~A. Let z=x+iy be the natural coordinate in D and w=u+iv the natural coordinate in B, we have

area(A)=fAdXdy=fBIcP'(w)12dudv=foh(follqY(w)12du)dv

>I I '(w)ldu do >- h,

where the In'st inequality is Schwarz's inequality, and the second comes from the isoperimetric inequality: any closed curve which surrounds K has length >I 1.

This can be rewritten area(D)-area(K)~mod(A) when area(K)= 1/4zt, or

area(D)- area(K) I> 4~r mod(A). area(K)

In this form, both sides are scale-independent, proving the proposition. Q.E.D. for Proposition 5. I0

Remark 5.11. (a) The inequality in Proposition 5.10

can be improved to give

area(D) ~> area(K) (I +4~r mod(A))

area(D) t> area(K) e 4~m~

For each n divide the annulus A into a nested sequence of n annuli, all of modulus mod(A)/n. For each such annulus Ai let Di be the disc bounded by the outer boundary of Ai. We obtain a sequence of compact sets K=KocKl=Dlc...cKn_l=D~_ 1 in D~=D with

area(Di+ 1) I> area(K/) (1 +4~tmod(A)/n) for i = 0, 1 .. . . . n - 1.

Therefore area(D)~area(K)(l+4~rmod(A)/n) n for all n and the improved inequality follows.

(b) One can prove that area(D)=area(K) e 4'~m~ if and only if K and D are exact concentric discs. (It does not follow from the proof given above.)


Now consider a cubic polynomial with one critical point escaping and the other not, and with the nests of all ends of C-Ke divergent. Let MN be the set of annuli at level N, and for each such annulus A, let Da be the disc bounded by its outer boundary. For each A E MN, let u(A) E MlV_ 1 be the annulus at level N - I such that AcDuta) and set

area(D A) /z(A) =

2 area(DA') A' Eu-I(A)

Define v(A) inductively by

v(A) = v(u(A))#(A),

starting the induction by v(Ao)=lt(Ao) where A0 is the unique element of M0. Let CN=infaea, v(A); for any A E MN we have v(u(A))/~(A)>-cN.

LEMMA 5.12. The sequence of numbers Co, cl, c2 . . . . tends to oo.

Proof. The opposite statement is that there exists a constant c such that for all N there is a product as above which is smaller than c. Since the sets Ms are finite, we can choose by a diagonal argument a sequence of annuli Ai all belonging to the same nest, and such that the infinite product l-I/~(Ai)<~c. In that case there exists a constant C such that

and then

log#(Ai) >/C(p(A3- I) for all i,

log c ~> Z log,u(Ai) >~ C Z (l't(Ai)- 1) I> 4~tC 2 mod(Ai)

which contradicts the hypothesis that all nests are divergent. Q.E.D. for Lemma 5.12.

Proof of Theorem 5.9. First observe

Z r(u(a'))area(Da')= 2 v(a) Z area(Da') A ' E ,~t N AEMIw I A ' E u - I ( A )

Z v(A) area(Da)[l~(A) A E.Y~N_ 1

2 v(u(A))area(Da)" AE~N_t

THE MONODROMY OF PATrERNS 281

Hence E v(u(A'))area(DA, ) is a constant C independent of N. Moreover

area(Da') = E area (DA)/~(A) A' E .~ N A E S4IN_ 1

<.c~11 ~ v(u(A))area(DA) A e ~N- !

= CCN'-r Therefore E area(DA) tends to 0 as N tends to oo by Lemma 5.12.

Q.E 9 for Theorem 5 9

Remark 5.13. This proof can be adapted to show that the Julia set of many other polynomials P have measure 0 when they are Cantor sets. The only information we need is that the critical level curves of hp define divergent nests. This occurs for instance in any degree when all critical points escape. The result is known in that case (see for instance [DHI] p. 40), but the previous proof is quite delicate. This proof appears to be simpler, since in this case the moduli of the annuli of nests are bounded below 9

We will see in Chapter 12 some other examples of Julia sets of measure 0 in higher degrees; they are cases where the tableau argument applies 9

Chapter 6. Monodromy of patterns

6.1. The local trivialization of the pattern bundle

The bundle PN: UN--->C-D defined in Section 2.8 is of course not analytically locally trivial, since we have seen in Proposition 2.9 that only two different patterns are isomorphic.

On the other hand, PN: UN-->C-[) is topologically locally trivial, and we will now proceed to give an explicit local trivialization. The construction is essentially an extension of the "wring construction" of [BH].

In Section 8 of that paper we defined a family of complex structures cr u on C - / ) , with corresponding Beltrami forms/zu(z), parametrized by u=s+it in the right half- plane s>0 and given by the formula

u - l s dz ~ u ( z ) = - - u+lz dz


282 B. BRANNER AND J, H. HUBBARD

The map f~ :C- /%-+C- / ) giving by z~z lz] ~-~ is a diffeomorphism commuting with z ~ z ~ for any n>0, and satisfies

& 8f. =P"'

i.e. it integrates the complex structure o.. Take a Riemann surface R E ~N(r and consider the complex structure on RN

obtained as follows. Let

R.= {zERIhR(z)> l~ ]r } 3" ;

on R0 = C-/~1r [ put the structure a., then on R1 put &*(a.), etc. Call the resulting Riemann surfaces (R., o.), so (2~, o.)=(R N, o.). Since (R, o.) is obtained from (RI, o.) by a succession of standard triple covers, exactly as any pattern, it must be uniquely isomorphic to a unique element R(u)E~N(~') for r162162 "-~. Call the isomorphism above fR.: (R, o.)---~R(u); clearly it extends fu on R0.

Remark 6.1. Let RE~N(~). For each O<~n<~N the isomorphism fR,. maps the critical annulus C.(R) onto the critical annulus C.(R(u)) and for each O<.n<N the critical value annulus B._I(R) onto the critical value annulus B._~(R(u)). Moreover for any annulus A of R nested inside an annulus A' the annulus fR.(A) of R(u) is nested inside the annulus fR.(A').

Choose r and choose U a neighborhood of 1 in the right half-plane so that the mapping U---~C-/) given by u ~ l ~ l ~-' is injective; let V be its image. Set P N I ( ( t ~ } ) = U N (~).

PROPOSITION 6.2. (a) The mapping UN(~)X U--*p~I(V) defined by (z,u)~-~fR,,(z) for z ER E ~N(~) and u E U is a homeomorphism, and the diagram

fR,u UN(r U -- p~,~(V)

U = V

commutes. In particular, PN: UN---~C-L) is topologically a locally trivial fibration.

THE MONODROMY OF PATTERNS 283

(b) I f V1 and V2 are two simply connected open subsets o f C-15 with V1 f)V2 connected and ~E Vl f)V2, then the local trivializations given in (a) coincide above Vl n V2.

6.2. The monodromy of the pattern bundle

A locally trivial bundle over a circle, and hence also over C-/), is topologically determined by the isotopy class of its monodromy. In particular, the locally trivial bundle PN: UN-~C-19 is determined by its monodromy, which is a homeomorphism of UN (~), only defined up to isotopy.

Proposition 6.2(b) says that the local trivializations induced by wringing the complex structure fit together to give a trivialization of the pull back ofpN: UN-->C-/) to the universal cover of C-/), hence computes this monodromy: it is the map which sends zERE~N(~) to fg,u(z)ER(u), when ~l~lU-l=~, i.e. when u(~)=l+2:~i/!og 1~1. More precisely, choose some r > l as base point in C-/), to get:

THEOREM 6.3. (a) For all n~O there exists unique homeomorphisms m,: U,(r)--> U,(r) such that the diagrams

m n U.(r) = U.(r)

~n ~n

U.-l(r) m_, = U.-i(r)

commute, and m0: Uo(r)--->Uo(r) is the map f~: C--l)r--->C--[)rfOr

u = 1 +2zti/log r.

(b) The monodromy mN: Ulv(r)---~Ug(r) o f the bundle PN: UN---~C-15 induced by the canonical local trivializations above is the mapping mlv.

Proof. Going along the vertical line u=l+it and starting with any R E UN(r), the path t,-*pN(R(u(O))=r ~(0 in C-L) returns to r for the first time for u=l+2~ri/log r, and forms a loop which generates Jtl(C-/)). So fu with this value of u is the monodromy of the bundle PN: UN---~C-/)" Q.E.D. for Theorem 6.3


Let R ~ ~N(r) and let A be any annulus of R at level O<~n<.N. Define the period of A under the monodromy mN to be the smallest k such that m~vk(A)=A. It follows from Remark 6.1 and Theorem 6.3 that the period under mN of the critical annuli C.(R) and the critical value annuli B._~(R) is the same number. Moreover any other annulus A of R has a period under mN which is a multiple of that.

PROPOSITION 6.4. (a) Let R E ~N(r). The periods under the monodromy mN of a nested sequence of annuli Ao, Al . . . . . AN of R form an increasing sequence of powers of 2.

(b) Let R ~ ~(r ) . Suppose the critical end is periodic of period k and level no; i.e. ~k(CN(R))=CN_k(R) for all N>>-no and ~r~J(CN(R))4=CN_j(R) for 0<j<k. Then for all N>~no the period of CN(R) and BN_I(R) under the monodromy mN is equal to the period of C~o(R) under the monodromy m~o.

Proof. (a) If an annulus A is nested inside A' then the period of A under mN is of course a multiple of the period of A', so we only need to prove that the period of any annulus A of R at level O<~n<~N is a power of 2. The proof is by induction on N.

For N= 1 there is a unique pattern RI E ~l(r). The period under ml of the annulus at level 0 and the two annuli at level l equals 1.

Suppose the statement is true for any pattern in ~N_~(r). We shall prove it true for any R E ~ r ) .

From ~rN(CN(R))=iRI(BN_1(R)) in SN(R) and ~rNomN=mN_l O~r N we conclude that the period of CN(R) under mN equals the period of i~I(BN_I(R)) under raN_ r From the induction hypothesis we know that the period of i~I(BN_I(R)) under mN_ i is a power of 2. Denote this period by k(C~R)).

Let A be any annulus of R different from CN(R) at level O<n<<.N, and let A' denote the annulus at level n -1 in which A is nested. The annulus erN(A)=A in SN(R) is nested inside ~rN(A')=A'. Let k(A) and k(i~I(A')) denote the period of A and i~1(A ') under mN_ 1 respectively. The number

k = sup(k(CN(R)), k(A), k(i~'(A')))

is a power of 2. Using JrNomN=mN_t oaz N we see that k is the smallest number such that the annulus m~k(A) is nested inside A' and mapped to ,/~ under Jru. If A'4:C,_~(R) then the annulus A is the only one with this property. If A'=C,_~(R) then there are two annuli with this property. If m~k(A)*A then the period of A under mN equals 2k, in all other cases it equals k.

(b) This follows immediately by inspection from the proof of (a). Q.E.D. for Proposition 6.4

THE MONODROMY OF PATTERNS 285

6.3. The m o n o d r o m y of the quotient pattern bundle

The monodromy of the quotient bundle PN: UN~C-19 described in Section 2.9 is a "functional square root" of the mapping mN above, and actually rather more natural than mN.

We chose r> 1 as base point in C-/5 for the bundle PN: UN-'->C-19, so it is natural to choose ? E C-/9 as the base point for PN: UN--~C-19 since UN (r) is naturally isomorphic to 0N (r2).

THEOREM 6.5. (a) For all n>-O there exist unique homeomorphisms rhn: UN(r)---> UN(r) such that the diagrams

U.(r)

Un-l(r)

: U,(r)

~n

mn- 1 - U . _ # )

commute, and rh0: Uo(r)--->Uo(r) is the m a p C-l~r"">C-lD r given by z~-->-fu(z) for

u = 1 +~ri/log r.

(b) The monodromy rhN: (-IN(r2)--*ON(r 2) o f the bundle PN: 0N--~C--19 is the mapping rhN.

(c) The monodromy mN of pN: UN-->C-D is given by (rhN) ~

Proof. Going along the vertical line u=l+it and starting with any R E 0N(r2), the path t~pn(R(u(t)))=r 2u(t) in C-/9 returns to r 2 for the first time for u= l+~ri/log r, and forms a loop which generates ~q(C-D). So -fu with this value of u is the monodromy of the bundle PN: ~-]N '''>C-/~. Q.E.D. for Theorem 6.5

As for the monodromy ms we can define the period of an annulus A of a pattern R under the monodromy r~N. We get a proposition analogous to Proposition 6.4.

PROPOSITION 6.6. (a) Let R E ~N(r). The periods under the monodromy rhN of a nested sequence o f annuli Ao, A1 . . . . . AN of R form an increasing sequence o f powers of 2.

(b) Suppose R E ~| and the critical end is periodic o f period k and level no. Then for all N>-no the period Of CN (R) and BN_I(R) under the monodromy thN is equal to the period o f Cno(R) under rh~o.


6.4. Fractional Dehn twists

It is possible to be more explicit about the mapping rhN, in terms of fractional Dehn twists.

Suppose we are given annuli A and A' of the same modulus M, both with an "inner" and an "outer" boundary. By the standard conformal representation of annuli onto regions

{Z[ e -~M ~< Izl ~< 1 },

we can consider both outer boundaries as metric circles. Suppose we are given an orientation preserving isometryffrom the outer boundary of A to the outer boundary of A'. The map f can be extended to the fractional Dehn twist of fraction q as follows: identify both A and A' to the standard annulus as above so that fbecomes the identity, and consider the map

Qe i~ ~ Qe i(O+q(l~

See Figure 6.1. The definition makes sense for any real number q, but in the following q will always be of the form 2 -k for some k~>0.

We shall assign to each annulus A of each R E ~N(~) the fraction of A

q(A) = 2:r rood(A) = rood(A) . log Ir mod(C0)

q-- 1/3

Fig. 6.1

287

(a)

THE MONODROMY OF PATTERNS

Fig. 6.2 (b)

This can also be done recursively by setting q(A)= I for the unique annulus Co of R, and then for any annulus A by setting

q(~tR(A)) q(A ) -- deg(~rR]a ).

TrIEOR~M 6.7. The monodromy rhN of the bundle 13N: (]N---~C-E) can be defined recursively as follows:

(a) On (]o(r 2) the map rho is the map C-/)r '--~C-Dr given by z~-->-fu(z) for u= 1 +ari/log r. More explicitely, it maps z=Qe iO to -z[zl u - l= -Qe i~~176176

(b) Any pattern R E ~n+l(r) has a critical value annulus Bn(R); R is mapped to the R' corresponding to the critical value annulus Bn(R')=th,(Bn(R)); if A is an annulus o f R-iR(sn+l(R)) with outer boundary y then A is mapped to the annulus A' o f R' with outer boundary Y'=mn(Y) by the extension o f rhnly as a fractional Dehn twist o f fraction q(A).

Proof. Statement (a) is already included in Theorem 6.5. Statement (b) is proved by induction on n. The mapping rh.+~ must make the following diagram commute:

A

zR(A)

m n + l

ffl n

~- A'

z%(a ')


Fig. 6.3

If r~n: ~rR(A)---~ztR,(A' ) is a fractional Dehn twist of fraction q, then rhn+l:A---~A' must be a fractional Dehn twist of fraction q/deg0tR). The numbers q(A) were defined so as to have this property. Q.E.D. for Theorem 6.7

Figure 6.2 shows the images ??12(a ) of the annuli at level 2 for both patterns R E ~2(r). In both cases the annuli a l and A2 are interchanged. Figure 6.3 shows the images th3(A) of the annuli at level 3 for a particular R E ~3(r).

Chapter 7. Parapatterns

Our next object is to describe the parameter space for patterns. As it turns out, practically all the work has already been done.

7.1. Parapatterns for fixed

Each pattern RE ~N(~), N>0, has a deepest critical annulus CN(R), and a deepest critical value annulus BN_I(R ). The ~-parapattern is defined as the quotient of the disjoint union

U BN_,(R---~)]/-- with the equivalence identifying for each N > I and for each R E ~N (C), the point x in the outer boundary of BN(SN(R)) tO iR(X) in the inner boundary of BN_I(R). It is quite

PARAPATTERNS 289

possible to give this space the structure of a Riemann surface directly, but the gluing takes place on a graph and this makes the construction a little fussy. There is an alternative equivalent definition where all the gluing takes place on open sets.

Each R contains the sub-Riemann surface V(R) consisting of the annulus BN_x(R) and all points inside it, and V'(R)= V(R)-B N_ t(R), which is a disjoint union of annuli as defined in Section 2.8. Recall that ~1(~) has a unique element R1. We can also define fl(~) to be the union

s O 0 V ( R ) ] / ~ N>I RE~N(~)

with the equivalence identifying, for each N > I and for each R E ~N(~), the point xE V'(SN(R)) to iN(x) E V(R).

With this definition the space f~(~) has a natural structure as a Riemann surface with boundary. The boundary is the outer boundary of Bo(R1).

7.2. The potential function H, critical graphs and arguments

Any parapattern naturally comes with a harmonic mapping H=H(~): f~(~)---~R+ with

H( ~)IBN_,(R) = h R]B N_ 1(1r The r of depth N>~O is the subset

~2N(~) = { (~, x) E ff~( ~) l ~ < H(~,x) ~ 31og '~' }.

The critical values of H are the numbers log 1~1/3", for n~>0. Let

Q(r log r FN(~)= (~,x)E H(~,x)= 3-~-_~ j for n~>0.

Arguments of points (~, x)E f~(~) and arguments of graphs are defined analogously to arguments of points in patterns in Section 2.4.

7.3. The real parapattern

Choose r> l real. Figure 7.1 represents ~3(r) (cf. Figure 2.5 of the tree of real patterns down to level 3).

The following proposition is the analogue of Proposition 2.6.

PROPOSITION 7.1. Each critical point x of the potential function H=H(r): ~(r)---~R+ satisfies H(x)=logr/3 k-1 for some k~l. It has precisely two ascending rays,

1/3

B. BRANNER AND J. H. HUBBARD 290

Fig. 7. l

which intersect the boundary F0(r) of Q(r) at points with arguments o f the form p/3k for some p not divisible by 3. Moreover all such rational arguments are obtained this way.

Proof. The proof of course goes by induction. Start the induction by observing that there are precisely two ascending rays emanating from the unique critical point of H in Fl(r) with arguments 1/3 and 2/3.

The inductive hypothesis says that the result is true for all k<N. Take any argument of the form p/3 N with (p,3)= I; using Proposition 2.6 we get a unique pattern R E ~N(r) for which p/3NE arg(F~R)). The descending ray of that argument leads to a critical point x E F~(R), from which exactly one other ascending ray emanates, and leads to a point of argument p'/3 N for some p' not divisible by 3. The point x belongs to V(R), which is contained in ~(r). Q.E.D. for Proposition 7.1

7.4. Parapattern isomorphisms

Let T(r ~(r be the mapping defined by

T(OI~(R~ = rRIv(R~

where ~R is the involution from Proposition 2.9. This mapping T is well defined because rR is compatible with iR.

PARAPATTERNS 291

PROPOSITION 7.2. The mapping T(0: ~(~)--~f2(-~) is an analytic isomorphism. I f F: fl(O-->f2(r/) is an analytic isomorphism, then either rl= ~ and F=id or r l=-~ and F=T.

Proof. The mapping T is analytic because its restriction to each V(R) is analytic. The potential H on a parapattern can be reconstructed as follows. Suppose W is a

parapattern. Consider the unique harmonic function h on W taking the value 1 on the (outer) boundary of W and the value 0 at the ends. This function differs (a posteriori) from H by an multiplicative constant. The constant in question can be determined from the moduli Ml and M2 of the two annuli between h-l(a) and h-l(b), where a and b are the two highest critical values of h. It will always be true (still a posteriori) that up to numbering M~=2M2, and then

H = 3:rMlh.

We know l~2l since log ]~21=2~M~. We can reconstruct the argument of ~2 exactly as we did for a pattern in Remark 2.11. So W equals either fl(O or f ~ ( - 0 for some ~ and it is not possible to distinguish them, since they are analytically isomorphic as explained above. Q.E.D. for Proposition 7.2

7.5. The parapattern bundle

The entire construction of a parapattern is functorial, which means that we can put parameters into the construction.

THEOREM 7.3. (a) There exists a unique topology on ~ = U ~ ( 0 making it a manifold with boundary and a unique structure o f a 2-dimensional complex manifold on the interior such that the canonical map VN---~(2 is an analytic chart for every N ~ I.

(b) With this structure, the projection

p: ~--> C-f)

given by 1~( ~, x)=~ for ( ~, x)6 int(~(O) is an analytic submersion.

Proof. The canonical maps VN---~(~ are injective: the equivalence relation never identifies different points of UN. The charts above clearly cover the interior of ~ . Since the equivalence is induced by the maps iN which are analytic by Proposition 2.13, these charts define a complex structure. Q.E.D. for Theorem 7.3


7.6. The quotient parapattern bundle

PROPOSITION 7.4. The isomorphisms T(~): f](r defined in Proposition 7.2 define an analytic involution of the bundle (2 covering the involution ~ - ~ of C-f).

The proof is left to the reader. Define the analytic manifold with boundary ~=~/T; the mapping p([z])=(p(z)) 2 is

an analytic submersion.

7.7. The bundle of clover leaves

We define an equivalence relation on ~) as follows: (~l,x)E~(r and (~2, y ) ~ " ] ( ~ 2 ) a re equivalent if and only

~13~---~2 3 and x=~l 3, y----~2 3.

The quotient space

if (r or

with the quotient topology is a manifold with a "ramified boundary" 8 f~=O~/ - (see the remark after Proposition 12.9 in [BH]).

This requires a bit of explanation: the boundary of fl(r is the circle z=l l 3, so (r ~3) is a point of the boundary of f~(r Thus the only gluing taking place is to identify the point (r ~3) E oa(fl(~)) to the points (j~, ~3) E 0(fl(jr and (j2~, ~3) E 0(fl(j2~)), where j=e 2~i/3. In particular the map

p: f ~ C-/)

given by (r z)~-~r 3 is well defined, and its fiber above r is f~(r U f~(jr 0 f~(fr glued to each other by identifying one point of each boundary. (This is closely related to the content of Remark 3.4.)

Chapter 8. Parapatterns and polynomials

In this chapter we shall show that every pattern of finite depth can be realized by a polynomial. This will allow us to identify the escape locus, i.e. the part of parameter space for cubic polynomials in which both critical points escape, with the parapattern bundle.

PARAPA~TERNS AND POLYNOMIALS 293

8.1. The universality of parapatterns

Now suppose that A is an analytic manifold, and that (Pa)ae,x is a family of monic cubic polynomials depending analytically on A and with labelled critical points o~(2), to2(2) being analytic functions of the parameter.

If for all 2 EA, we have ha(~Ol(2))>h~(to2(2))>0, then by Corollary 3.3 there exists a unique ~(2), a unique N(2), a unique pattern R(2)~ ~N<~)+1(~(2)) and an isomorphism q3~: U~(to2(2))--~R(2 ) which is defined at the critical value Pa(to2(2)). Recall from Section 2.8 the subsets V'(R(2))~V(R(2))~R(2). Clearly

A(2) =

is in the V(R(2))-V'(R(2)). Using the Remark 3.4, the construction above still goes through if ha(toz(2))=

hx(~o2(2)) and oJl(2)~to2(2) , and in case tol(2)=to2(2), the identification we have made on (2 still makes the mapping ~a: A--->Q well defined.

PROPOSIa'ION 8.1. Let A be an analytic manifold with boundary and (Pa)z~A an analytic family o f monic cubic polynomials with labelled critical points ~o1(2), to2(2) depending analytically on 2 E int(A), and satisfying h~(tol)~>h~(a~2)>0. Then the mapping qbA: A---->Q is continuous and analytic on int A.

Proof. If the family (Pa)a~A is constant, the result is obvious. Otherwise, if 20Eint(A), then ePA(20)Eintfl since h~(col)-h~(to 2) is harmonic on int(A). Set depth(P~o) = N(20) = N. There exists a neighborhood A ' c A of 20 such that, for all 2 E A', we have N<~N(2)<<_N+I and q0~(Px(o~2(2))) lies in VN+I; since VN+ 1 is a chart of ~ it follows that the mapping 9~, is continuous at 20 and analytic in a neighborhood of 2o if 20 E int A.

If 20 fi 8A and dPA(20) 6 aff~ then the two critical points tOl(20), to2(20) escape at the same rate. There exists a neighborhood A ' c A of 20 such that, for all 2EA' , we have N(2)=0. In this case Pz(t92(2)) is in the domain of q~. Q.E.D. for Proposition 8.1

8.2. Parapatterns and the escape locus

Now let us apply Proposition 8.1 to a particular A. But first we need some notation. We shall parametrize cubic polynomials by

Pa. b(z) = z3-3a2z+b"

294 B. BRANNER AND J. H . HUBBARD

The critical points ofPa, b are +a. Let 5ebe the set of parameters for which at least one critical point escapes to infinity; i.e.

6e= ((a, b) E C21 sup (ha, b(--a), ha, b(a)) > 0}.

This set naturally splits into two symmetrical sets Se= b ~ U 5e- where

6~ + = {(a, b) E 5e I 0 < ha, b(-a) ~< ha, b(a)}.

The labelled critical points of Pa, b for (a, b)E 5P + are wl(a, b)=a and co2(a, b ) = - a , and the co-critical point w~'(a, b) is - 2a .

Proposition 1 1.3 and Corollary 13.3 in [BH] state that the mapping ~p+: O~ given by ~0+(a, b)=q0a, b(Pa, b(a)) is a non-trivial fiber bundle with fibers homeomorphic to a trefoil clover leaf, i.e. three closed discs with one boundary point in common. Moreover this corollary implies that the mapping

73+: 5~+-ase+--* C-/)

given by 73+(a, b)=limz_~_za q~a,b(z) is a non trivial fiber bundle with fibers homeomorphic to D. Note that - 2 a is the co-critical point of a so that 73+ is a cube root of ~0 + which is well defined as 10ng as the locus where the critical points coincide has been removed from the domain. (See Remark 3.4.) We shall denote the fiber above r by

~e+( r = ( 7 3 + ) - 1 ( 0 ,

the r the trefoil clover. In this section we shall only consider a subset of 5e: the escape locus ~ which is

the set of parameters for which both critical points escape to infinity; i.e.

= ((a, b)l inf(ha, b(--a), ha, b(a)) > 0).

Let ~ +=5P+ n ~ and ~ +(r162 ~. In Chapter 10 we shall consider the complemen- tary subset ~=Se-~, i.e., the set of parameters for which one critical point escapes to infinity and the other critical point has bounded orbit. Let ~+=Se+fl ~ and ~+(r ~+(~) N ~. We have 5e= ~U ~, 5e + = ~ + U ~+ and ~+(~)= ~ +(~) U ~+(~).

Now we are ready to apply Proposition 8.1 to A= ~+.

TrlEO~Er~ 8.2. The mapping ~ + : ~g+--,Q is a homeomorphism, analytic in the interior.

PARAPATrERNS AND POLYNOMIALS 295

Proof. We have seen in Proposition 8.1 that *~+ is analytic in the interior of ~ + - a ~ + ; we will now show that it is proper of degree one. Let P , fi ~+ be a sequence of polynomials with the sequence z.=~,+(P,) convergent in ~ . By using the local triviality of the bundle V+: O~ and the compactness of each fiber we can assume that the P , converge to a polynomial P. Since the sequence (z,) is convergent, the sequence lies in ~ u for some fixed N, so we have the bounds

1 ha, b(-a) - -<~ <~1, 3 N ha, b(a)

which are still true of P; so P is in ~+. This shows that the map is proper, hence it has a degree. We showed in [BH], Theorem 12.2 that this degree is I.

Q.E.D. for Theorem 8.2

Remark 8.3. Theorem 8.2 says that given RE ~N(~) and x E V ( R ) - V ' ( R ) there exists a unique polynomial PE ~+ so that cbe(P(-a))=x. This shows that every ~- pattern can be realized by a polynomial. Moreover, it identifies the analytic structure of int ~+ and that of int ~ . A more precise statement is found in Corollary 8.4 below.

Since the mapping/~: ~--->C-/) is not well defined at the equivalence classes we define the mapping p: g2--->C-/5 to be p(x)=(#(x)) 3 if x E Int(g2) and p(x)= r if x E a~(~). This is well defined (even at the gluing point).

COROLLARY 8.4. (a) The mapping

r tg +

C - / )

is a bundle isomorphism. (b) The mapping (~+1~+r g2(~)--~+(~) is analytic in the interior.

Remark 8.5. It follows that the mapping p: f~--+C-/) is a non trivial fiber bundle. Moreover we see that any parapattern g2(~) is realized in the parameter space of cubic polynomials.


Chapter 9. Ends of parapatterns and polynomials of infinite depth

The parapattern space f~(~) has ends, E(ff~(~)), which correspond bijectively to the connected components of

~+(~) = ~ + ( 0 - ~ +(~).

The set of connected components of ~+(~) with the quotient topology is a Cantor set homeomorphic to E(f2(~)).

We will show in this chapter that countably many of these components are homeomorphic to the Mandelbrot set M and that all other components are points; the dichotomy will correspond precisely to the dichotomy in ~=(~) between patterns with periodic critical ends and non-periodic critical ends which we encountered in Chapters 4 and 5.

9.1. The two types of ends

The appearance of Mandelbrot sets in the parameter space for cubic polynomials is explained by families of polynomial-like mappings of degree 2. We saw in Chapter 5 that some restrictions of iterates of cubic polynomials are polynomial-like of degree 2, and in this section we will consider the subsets of parameter space in which this occurs. We will see that this gives rise to infinitely many Mandelbrot-like families.

Each end of x 6 E(Q(~)) corresponds to a pattern

pat(x) = R fi ~|

Indeed, such an end corresponds to a sequence of patterns (Ro, R1,Rz .... ) and the sequence (Bo(RI),Bi(R2) . . . . ) of critical value annuli. Conversely, to any pattern R=(Ro, RI,R2 . . . . ) in ~|162 we can associate the sequence of nested annuli (B0(Ri), BI(R 2) .... ), which surrounds a unique end in E(Q(~)). So pat: E(ff~(~))-+~| establishes a one-to-one correspondence between E(f~(~)) and ~ (~) .

THEOREM 9.1. (a) I f the critical end o f pat(x) is periodic then the component Mx o f ~+(~)--~L~'+(~) - $+(~) corresponding to x is homeomorphic to the Mandelbrot set M.

(b) I f the critical end o f pat(x) is not periodic then the component o f ~+(~)==LP+(~)-$+(~) corresponding to x is a point.

Proof of(b). Let us first prove (b), which is easy with the main result of Chapter 4. The annuli of the nest of an end x are precisely the same as the annuli of the nest of the

ENDS OF PARAPATrERNS 297

critical value end of R=pat(x). If the critical end of R is not periodic, then the modulus of this end is infinite (Theorem 4.3), so the modulus of the end x is infinite, and the component of ~+(~)-~f+(~) corresponding to x must be a point (Propositions 5.4 and 5.5). Q.E.D. for Theorem 9.1 (b)

9.2. Mandelbrot-like families in cubic polynomials

We shall first recall the notion of Mandelbrot-like families of polynomial-like mappings [DH2] since that is needed for the proof of (a).

Let f = (J~: U~--> Uz)x e A be an analytic family of polynomial-like mappings of degree 2 with A homeomorphic to a disc. Let cox denote the critical point ofj~. Set

Mf = (;t E AI Kfa connected}.

It follows from the straightening theorem, quoted in Section 5.1, that there is a map

~: My--~ M

defined by]~, A E Mf, is hybrid equivalent to Qztz)(z)=zE+z(2). The family f=(fz),IEA is called Mandelbrot-like if Z is a homeomorphism.

i

We shall need Proposition 21 from [DH2] in the following form:

PROPOSITION. Let f=(fz: U~--~U~)2EA be an analytic family o f polynomial-like mappings o f degree 2 with A homeomorphic to D. Suppose there exists a subset A ~ A homeomorphic to D such that f~(toz) E Uz- U'~ for A E A-int(A). The map Z: Mr---~M is a homeomorphism if f~(ogz)-w~ turns around 0 once as 2 describes aA.

Now suppose x EE(ff2(~)),R=(Ro, RI,R2 .... )=pat(x) and the critical end c of R is periodic of some period k and some level no (cf. Section 5.1)

As recalled in Chapter 8 it follows from Theorem 13.2 and Corollary 13.3 in [BH] that Sg+(~) is homeomorphic to a disc. The critical value annulus Bno_l(R) is naturally embedded in ff2(~) and the outer boundary of ~l+(Bno_l(R)) bounds a region Ax homeomorphic to an open disc. All the polynomials which belong to Ax have patterns which to depth no coincide with Rn0. The corresponding regions of the dynamical planes can therefore be identified through tP,,b (Theorem 3.1).

Example 9.2. For each ~ there are unique patterns R I , R 2 ~ ( ~ ) for which the critical ends Cl, c2 have period 1 and 2 respectively. The critical value end is bi=~rR,(ci)



Fig. 9.1

for i= I, 2. Figure 9.1 shows the regions Abl and Ab~ corresponding to these ends. Figure 5.4. illustrates any polynomial in Ab =~+(r these are the "polynomial-like mappings which stand out particularly" mentioned in the introduction to Chapter 5. Figure 5.6 illustrates any polynomial in Ab.

Recall from Section 5.1 the definition of Wn as the smallest simply-connected domain in C containing the critical annulus Cn(P). In Chapter 5 they are only considered for polynomials of infinite depth, but the definition clearly makes sense even if P has finiteodepth so long as n~>depth(P).

PROPOSITION 9.3. Let x EE(~(~)). Suppose the critical end o f pat(x) is periodic o f period k and level no.

(a) For all (a, b)EAx, the restriction o f the k-th iterate Of Pa, b tO Wa, b,~o:

f a , ~ o k b - P,~,blWo. b,,o" W~,b,.o-'-> W~,b,<.o-k)

is polynomial-like o f degree 2, where W~, b, ~ is the smallest simply-connected domain in C containing C~(Pa, b).

(b) A polynomial P~, b with (a, b)E Ax belongs to Mx if and only i f the critical point o f the polynomial-like mapping fa, b does not escape.

(c) The component Mx is homeomorphic to the Mandelbrot set M.

Proof. (a) and Co) The polynomials P~, b for (a, b) E A~ satisfy the requirements of Proposition 5.7.

ENDS OF PARAPATrERNS 299

Fig. 9.2

Let

(C) This follows from Proposition 21 in [DH2] stated above. Choose any 0 with

log < o < log 3 n~ 3 n~ I "

Ax= ((a,b)EAx[HO~§ }, y = 0 A x and 6=~<k- l ) (~+(y) ) .

Note that the last formula above makes sense because ~§ lies in the annulus B~o_~(R) which is naturally a subset of both R and ~(~).

When (a, b) describes the boundary curve y, turning around Mx once, then ~a,b(Pa, b(--a)) describes the curve ~§ in Bno_l(R)cR. Since n~k-l) maps Bno_l(R) with degree 1 onto Cno_k(R) then ~a, b(P~ describes 6 turning around the critical

ok end c once. Hence the vector from - a to Pa, b(--a) turns around 0 once. We have exactly verified the hypotheses of the proposition above. Q.E.D. for Proposition 9.3

This ends the proof of (a) in Theorem 9.1. Q.E.D. for Theorem 9.1

Remark 9.4. Theorem 9.1 (b) says that given a pattern R E ~|162 with non-periodic critical end there exists a unique polynomial of infinite depth realizing R. Moreover, Theorem 9.1 (a) says that given a pattern R E ~| with periodic critical end there exist infinitely many polynomials of infinite depth realizing R.


(a)

Fig. 9.3

Recall from Section 8.2 that ~p+: 5e+--~C-/) given by ~+(a, b)=q)a,b(Pa, b(a)) is a non-trivial fiber bundle with fibers homeomorphic to a trefoil clover leaf. Figure 9.2 shows the fiber above r for some r> 1.

Figure 9.3 (a) shows a part of ~+(r) and Figure 9.3 (b) a blow up around Mbl in Abt. Figures 9.4 (a) and (b) show two successive blow ups of ~+(r) around Mb2 in Ab2.

(a)

Fig. 9.4

THE MONODROMY OF PARAPATrERNS

(a] n (b)

301

2/3 1/3

Fig. 9.5

Figures 9.5 (a) and (b) show parts of ~+(re 2~i/3) and .~+(re 4~i/3) respectively. The Julia sets shown in Figures 5.7, 5.8, 5.9 correspondto the polynomials chosen

to be the center of the Mandelbrot set Mb~ in Figures 9.3, 9.5 (a) and (b) respectively. The Julia set shown in Figure 5.10 corresponds to the polynomial at the center of the hyperbolic component of period 2 of the Mandelbrot set Mb, in ~+(r). The Julia set shown in Figure 5.11 corresponds to the polynomial at the center of the Mandelbrot set Mb2 in Af+(r). The Julia set shown in Figure 5.12 corresponds to the polynomial at the tip of the Mandelbrot set Mb~ in ~?+(r).

Chapter 10. The monodromy of parapatterns

Recall from Section 8.2 that ~p+: 5e+--->C-/) given by ~V+(a,b)=q~a,b(Pa, b(a)) is a non trivial fiber bundle with fibers homeomorphic to a trefoil clover leaf.

In Corollary 8.4 we have proved that qb+: ~+--->f~ is a bundle isomorphism. In this chapter we shall describe the non-triviality of the parapattern bundle

/~: O--->C-/) and the parapattern quotient bundle/~: O--->C-/) by describing their mono- drornies, All the work was done in Chapter 6; here we just collect results.

Moreover, the monodromies of the parapattern bundle and the quotient parapat-


tern bundle induce monodromies of the ends. We shall use this in Section 10.5 to describe the components of the subset ~ of parameter space for cubic polynomials with one critical point escaping to infinity and the other not.

10.1. The local trivialization of the parapattern bundle

The local trivializations

UN(~)x U - pN-~(V)

u ~ r162 ~ - , U .. V

described in Proposition 6.1 induce local trivializations of the subbundle

v , ~ , - - u N

\ / C-15.

Moreover, these local trivializations are compatible with the inclusions: the diagram

V N ( D x U V,NI(~~uiN /xid~/ U

" V,Av

--- v'~fv

u,--, ~1~1 ~-' - V

commutes. All of this is tedious to verify but essentially obvious, and left to the long- suffering reader.

As a result, the local trivializations of the pattern bundle given by wringing the complex structure induce local trivializations of the bundle #: ~---~C-/), and just as in Section 6.2 these local trivializations are canonical and define a global trivialization of the pullback of the bundle to the universal cover of C - / ) .

THE MONODROMY OF PARAPATTERNS 303

10.2. The monodromy of the parapattern bundle

As in Chapter 6 we choose some r > l as base point in C - / ) and describe the monodromy as a homeomorphism m: f~(r)--->f~(r).

Using the notation of Theorem 6.3, the discussion above can be summed up as follows:

THEOREM 10.1. The monodromy m: f~(r)-~(r) of the bundle #: (2-oC-19 induced by the canonical local trivialization above is given by

mIv(R)=mNIv(R) where R fi ~N (r).

Let A be any annulus at level N of Q(r). We define the period of A under the monodromy m to be the smallest k such that m~ Since each nested sequence of annuli in Q(r) is the nested sequence of critical value annuli (Bo(R), BI(R) . . . . ) for some R 6 ~| we get the following proposition as an immediately consequence of Proposi- tion 6.4.

PROPOSITION 10.2. (a) The periods under the monodromy m of a nested sequence of annuli Ao, A1 ..... AN .... of f~(r) form an increasing sequence of powers of 2.

(b) Let x6E(f~(r)). Suppose the critical end of R=pat(x)6 ~| is periodic. Then the periods under the monodromy m of the annuli in the nest of x are bounded.

10.3. The monodromy of the quotient parapattern bundle

The monodromy of the intermediate bundle

p: ~ - ->C- / )

is a "functional square root" of the mapping m above. We chose r > l as base point in C - / ) for the bundle #: ~---)C-/), so it is natural to

choose r2E C - / ) as the base point for p: 0- -~C- / ) since f2(r) is naturally isomorphic to g)(r2).

Using the notation of Theorem 6.5, we have:

THEOREM 10.3. (a) The monodromy rh: O(r2)---~(r 2) of the bundle #: (2-~C-19 is given by


a, lv: = ,~VN(R))

where R E ~N (r). (b) The monodromy m o f p: ~(r)--->C-/) is given by (K~) ~

PROPOSITION 10.4. (a) The periods under the monodromy rh o f a nested sequence o f annuli Ao, AI, . . . ,AN .. . . o f (2(rZ) form an increasing sequence o f powers o f 2.

(b) Let x EE(Q(r)). Suppose the critical end o f R=pat(x)6 ~| is periodic. Then the periods under the monodromy rh o f the annuli in the nest o f x are bounded.

Proofs. Again this is obvious and left to the reader.

10.4. Fractional Dehn twists

As in Chapter 6 it is possible to be more explicit about the monodromy in terms of fractional Dehn twists.

Recall that for each annulus A in f~(r) the fraction q(A) is defined by

q(A) = 2:t mod(A) _ mod(A) log r 2 mod(C0 ) "

The following theorem is an immediate consequence of Theorem 6.3 and Theorem 10.3.

THEOREM 10.5. The monodromy th: ~(r2)---->~)(r 2) of the bundle lJ: f2--->C-19 is given recursively as follows:

(a) On a~(r) the monodromy is the identity. (b) Suppose rh N_ 1 is given by induction. Let A be an annulus at level N with outer

boundary y. Then the mapping rhN is the extension o f rhN_ll ~ as a fractional Dehn twist o f fraction q(A).

In Figure 10.1 we label the annuli of Q(r) down to level 3 by the fraction of Dehn twist which occurs there. We shall make use of this in Sections 10.5 and 11.4.

10.5. Consequences for the polynomials of infinite depth

The monodromy described for the parapattern bundle induces a monodromy on the ends of f2(r), i.e., a homeomorphism

m: E(f~(r))---, E(Q(r)) .

THE MONODROMY OF PARAPATrERNS 305

1/2

, I

Fig. 10.1

We define a metric on the set of ends E(g2(r)) as follows:

1 d(x, y) = 3.+, ( for x, y E E(f2(r)),

where n is the smallest level at which there exists an annulus surrounding both x and y, or in other words if we have to climb to the critical graph F,+~ in order to go from x to y.

With this metric the set of ends E(~2(r)) forms a metric Cantor set and m: E(g2(r))---~E(g2(r)) is an isometry.

The orbit of an end x EE(ff2(r)) under m is defined to be (m~ z.

PROPOSITION 10.6. I f E is a metric Cantor set and m: E--~E an isometry, then the closure o f an orbit is a finite set or a Cantor set.

Proof. Suppose S=((m~ is an infinite orbit. In order to prove that X is a Cantor set we only have to prove that X is perfect. If not, there would exist e>0 and a E X such that d(a,x)>e for all x E X - { a } , i.e. d(a, m~ for all n:#0. Using that m is an isometry we get d(m~ m~ for all n4=k which contradicts the compactness of E. Q.E.D. for Proposit ion 10.6


An end xEE(f~(r)) is convergent if the modulus of the nest N(x) is finite and divergent otherwise. Recall that an end xEE(g2(r)) is convergent if and only if R=pat(x) E ~| has a critical end which is periodic under :rn. It follows from Proposi- tion 10.2 (b) that the period under m of a convergent end x is finite.

An end x E E(f~(r)) is therefore either (1) convergent, (2) divergent with finite orbit under m or (3) divergent with infinite orbit under m. We shall prove that these 3 types of ends of f~(r) correspond to 3 different types of

connected components of ~+=Se + - if + in the bundle ~+: ~+---~C-/5 where ~+(a, b)= q3a, b( -2a) (see Section 8.2). subbundle

where

For convenience let us discuss the components of the

~r§ " ~ r § -...-> S I

~r + = {(a, b) E ~+l ha.b(a) = log r} and ~+ ~- qga, b(-2a)/r.

In order to state the result we need the following definition: the dyadic solenoid is the projective limit

57 = lim (S I , z ~ z 2)

i.e., 5~ is the set of sequences ( .... z2, z~, zo) with zi§ and ]zo] = 1, endowed with the product topology.

A projective limit

!im (S 1, qn)

k n

with mappings qn of the form qn(z)=z z and infinitely many k~>0 is homeomorphic to the dyadic solenoid.

PROPOSITION 10.7. Any connected component in ~r + is homeomorphic to one o f the following 3 types:

(1) S l x M with the projection S 1 xM---~S 1 wrapping S I onto itself2 k times for some k>~O,

(2) S 1 with the projection SI--->S i wrapping S 1 onto itself2 k times for some k>-O, (3) the dyadic solenoid.

THE MONODROMY OF PARAPATrERNS

Fig. 10.2

307

Proof. Type 1 is a component containing Mxc~+(r) for some convergent end x E E(~(r)). Type 2 is a component containing a point component of ~+(r) corresponding to a divergent end with finite orbit under m. Type 3 is a component containing a point component of ~+(r) corresponding to a divergent end with infinite orbit under m. The proof is an immediate consequence of the Propositions 10.2 and 10.6, see also the schematic drawing in Figure 10.2. Q.E.D. for Proposition 10.7

Remark 10.8. (a) In type 1 the bundle SI• 1 c a n not be like a M6bius band since the Mandelbrot set must be twisted an integer number of times around itself before it closes up. I fx E E(f2(r)) is convergent and the period ofx under m is 2 k then the twisting is determined by 2 k mod N(x). Hence 2 k mod N(x) is an integer.

(b) In type 2 and conjecturally in type 1 the connected component is also an arcwise connected component. (Conjecturally refers to the local connectivity conjecture of the Mandelbrot set.) In type 3 the component is not arcwise connected, but the closure of any arcwise connected component.

(c) The dyadic solenoid also occurs in [HO], in the context of complex H6non mappings.

Let ~+(r denote the set of equivalence classes in ~+(~) under attine conjugation and set ~+=O~+(~). Anything said in this section about components in ~r + can be repeated for the bundle ~r+---~S l using the monodromy rh instead of m.

(a)

308


M

Ml

M2

Fig. 10.3

6 We shall give a sufficient condition for a divergent end x to correspond to a

component of type 2.

PROPOSITION 10.9. Let x 6E(g2(r)) and R=pat(x) 6 ~ ( r ) . Suppose the critical end o f R is non-recurrent. Then x has a finite orbit under lb.

Proof. Since the critical end of R is non-recurrent there exists an n~0 such that any annulus in the nest of x has a modulus equal to 2 k mod(C0) with O<.k<.n. It follows from Theorem 10.5 that the mapping rh ~ applied to any annulus in the nest o f x is a Dehn twist of integer twist, which maps each annulus in the nest of x to itself. For the induced monodromy on the ends we get rh~ Q.E.D. for Proposition I0.9

Remark 10.10. (a) Recall that polynomials with one critical point escaping to infinity and the other critical point strictly preperiodic (for instance Brolin's example) have non-recurrent critical tableaus.

(b) In Example 12.4 we shall define a critical marked grid called the Fibonacci grid with the critical end divergent and recurrent. The ends in E(g2(r which are realizations of the Fibonacci grid correspond to a component in ~,+ of type 3. We don't know if a critical marked grid with the critical end divergent and recurrent can give rise to a component of type 2.

THE FUNDAMENTAL GROUP

f o o o o / o o o

o o 1

309

Fig. 10.4

Polynomials belonging to the same component of ~+ have patterns which have the same critical marked grids. But there are polynomials not in the same component of 5~ + which nevertheless have the same marked grid.

Example 10.11. Figure 10.3 (a) and (b) show parts of ~+(r) for some r> l . The figure eights are the parts of F3(r) with arguments in [1/9, 2/9] and [7/9, 8/9] respectively. Four Mandelbrot-like sets Mi, i= 1,2,3, 4, are marked. It follows from Theorem 10.5 that M1 and M4 are contained in one component of ~+ and M2 and M3 in another. But they all have the same critical marked grid of period 5 which is shown in Figure 10.4.

Chapter 11. The fundamental group of the escape locus In this chapter we will compute the fundamental group of the escape locus. This is of interest because [BDK] have proved that the monodromy of Julia sets induces a homomorphism from this fundamental group to the group of automorphisms of the 1- sided 3-shift which is surjective.

There is really only one way to compute fundamental groups: van Kampen's theorem. Our computation is four successive applications of this result. The strategy is as follows.

Recall from Chapter 8 that the escape locus ~ naturally splits into two symmetrical sets ~= ~ + U ~ - where

~+ = {(a, b)[ 0 < ha, b(--a) <~ h,,0(a)};

moreover (Theorem 8.2) the mapping ~ + : ~g+---~f~ is a homeomorphism. The space is a quotient of (2 (7.2); and ~ is a fiber bundle over C - / ) , with fiber say fl(r), and monodromy computed in Theorem 10.1.


1/

?'(1/3, 2/3)

2/9

Po Po

7{ttg,~} Fig. 11.1

We will use van Kampen's theorem as follows: (1) Compute the fundamental group of f~(r), and the action ~u.'fft'l(~"~(r),e0) ---)

=l(fl(r), P0) of the monodromy on it. (2) Construct an isomorphism of the fundamental group of ~ with the semi-direct

product Z x~, ~tj(f~(r), P0). (3) The fundamental group of f~ is then given by the presentation (xt(~,Po),fl)

where fl is a new generator satisfying the relation fl3=a, where a is a particular element of xl((~,Po), representing the circle which is glued to itself.

(4) Finally, the fundamental group of ~= ~+ U ~ - is isomorphic to the amalgamated

s u m yt'l ( ~ +, P0)* rl(ff + t3 ~-, P0) "~I(~ - ' P0).

11.1. Choice of base point

First we need to pick a base point. We will choose any polynomial of the form po(z)=z3+b with b>2/V~. It is easy to show that such a polynomial belongs to the escape locus, in fact belongs to ~+ N ~- . Let H(Po)=logr>O. The polynomial p0 doesn't quite have a pattern, since it belongs to the identification locus under the projection g)---~f~; but the point P0=(r, r 3) 6 0f2(r) is one of the 3 points of ~ corresponding to this polynomial.

THE FUNDAMENTAL GROUP 311

11.2. The fundamental group of the real parapattern

Recall that the lobe fl(r), r> l , is drawn in Figure 7.1 and is described in Section 7.3. Proposition 7.1 defines an equivalence relation on Z[1/3]/Z, with all equivalence

classes containing precisely two elements, except the equivalence class of 0 which has only one element. Let X be the set of equivalence classes. For each class x= {pl/3 k, p2/3 k} fiX we can construct an element

Y~ f :h(Q(r), Po)

as follows: start at the base point, go counterclockwise around the boundary F0(r) until you first meet a point with argument in x, then follow the ray down to the critical point, and go back to F0(r) along the other ascending ray leading to the other dement of x, then return to the basepoint counterclockwise around the boundary (see Figure 11.1). This definition does not quite work for the element {0} of X; the element ~0 of ~h(fl(r),P0) corresponding to {0} is simply the boundary of f~(r), traversed once counterclockwise.

PROPOSITION 1 I. 1. The fundamental group :q(g2(r), P0) is the free group on the generators ~'x, x f X.

Proof. This is essentially the easiest case of Morse theory [M1]. First, if XN is the subset of X with representatives p/3 k with k<~N+ 1, then ~x is a

loop in ON(r) if XEXN. It is enough to prove that :rt(ON(r),Po) is the free group on y~, x E XN, since the fundamental group of an increasing union is the inductive limit of the fundamental groups; clearly the inductive limit of free groups on increasing finite sets of generators is the free group on the union.

If N=0, O0(r) is a deformation retract of the "figure eight" Fl(r), and ~tl(O0(r), P0) is the free group on the two generators ~0 and ~(1/3,2/3).

Suppose N>0; ifyi are the critical points of HwithH(yi)=(logr)/3 N, then ON_l(r)is a deformation retract of ON(r)-U{y~}. In fact, flow up along the gradient lines of H provides such a retraction, see [M1] for details.

Now apply van Kampen's theorem to the two subsets

Ul=ON(r)--U{Yi} and U2=Uri , iEXN--XN_,,

i.e. where the 7i are those Yx going through the yi. The intersection U~ N U2 can be deformed to the circle F0(r). The fundamental group of ON (r) is the amalgamated sum of the fundamental group of O N_ l(r), which is by induction the free group on generators


yx, xEXN_~, and of the fundamental group of Lly;, which is the free group with generators the Yi, i E XN--XN_ I and Y0, amalgamated over ~h(F0(r), P0) which is the free group on Y0.

The result follows from Lemma 11.2, applied to Z~=(XN-XlV_ 0 UX o and Z2=XlV_ ~. Q.E.D. for Proposition 11.1

For any set Z let Fz be the free group on Z.

LEMMA 11.2. diagram

is an amalgamated sum.

Let Z I , Z z be two sets; set Z0=zlnz2 and Z = Z I O Z 2. Then the

F J / F z '

The proof is left to the reader.

11.3. The fundamental group of the parapattern bundle

Let SI=R/Z, I=[0, 1] and x: I-->S ~ be the canonical map. Let Z be a topological space, and f : Z--,S l a locally trivial fibration, with pathwise connected fibers. Let Z0=f-l(0), and choose a base point z0 E Z0.

The pull back of f by ~r is trivial since I is contractible; a trivialization induces a mapping r making the diagram

commute, and which mr: Zo---,Zo is given by

IXZo - Z

, S I

is a homeomorphism on fibers. The induced monodromy

mr(z) = r(l, Z).


The mapping r can be chosen so that m,(zo)=Zo. In that case the map a: I--->Z given by t ~ r ( t , Zo) is a loop representing an element a EJh(Z, zo).

LEMMA 11.3. Let G=~q(Zo, Zo), let l~: G-->G be the map induced by mr and Z x # G be the semi-direct product. Then there is a unique isomorphism h: Zx~,G-->Jr1(Z, Zo) such that hIGx{l } is induced by the inclusion Zo-~Z, and hod, 0)=a.

Proof. This is classical: the long exact sequence of the fibration gives a short exact sequence

i 1 --~ G - * .7~I(Z, Zo) ---> Z---> 0;

and the map n ~ n a is a splitting, so that every element of ~rl(Z, zo) can be written uniquely as fig). (na). (A representative for the product x. y is given by first traversing a path for x and then a path for y.) So to prove the lemma, all we need to show is

i(g) " a = a. i(~t(g)).

Represent g by y: I--->Zo. Then the family of maps 6t: [0, 2]--->Z given by

f r(s, z0), 6t(s) = ~r(t , y ( s - t)),

I t ( s " I, zo),

l ~ s < ~ t t<~s<. t+l t+ l ~<s<~2

is a homotopy between paths representing i(g). a and a. i(/~(g)). Q.E.D. for Lemma 11.3

Lemma 11.3 can be applied to the fibration g)--->C-/). First, note that the monodromy m: ~2(r)--~g2(r) considered in Chapter 10 does satis~' m(Po)=Po. Let

/~: ~rl(g}(r), P0 )~ ~rl(g2(r), P0)

be the automorphism induced by m. Let a E:q(g),P0) be the class represented by the loop given by the trivialization in Chapter 10. Remember that the boundary of (2 is canonically the set of ((~,z)llzl=l~31}; the loop a is given by the parametrization t ~--~(re 2:~#, r3e 6~it ).

COROLLARY 11.4. There is a unique isomorphism Zxazh(ff2(r),Po)-->Jr~(O,Po) given by the canonical inclusion on arl(f~(r), Po) and sending (id, 0) to a.

21-928286 Acta Mathematica 169. [mprim~. le 10 novembre 1992


(a)

P0 P0

(c) (d)

Po Po

Fig. 11.2

11.4. Computing the monodromy of the parapattern bundle

We were a bit lighthearted above when we simply wrote "let

/z: :rl(f/(r), P0)--> :h(Q(r), P0)

be the automorphism induced by m". We do not know any simple way of computing the action of/z on the generators; in this section we will show how complicated this mapping/z actually is by computing a few examples. Note first that there is a homomorphism/~ induced by r~ such t h a t ~.~2=/2 , which is a bit simpler than/~ (without excess).


Our basic tool is the computation of rh in terms of fractional Dehn twists. Recall Figure 10.1 from Section 10.4 in which we labelled the annuli of f~(r) by the fraction of Dehn twist which occurs there. Using that figure, we obtain the following Figures 11.2 (a)-(d) for the images of Y~1/3,2/3}, )/{1/9,2/9}, )/{4/9,5/9} and )/(7/9, 8/9} .

We leave the diligent reader to verify that the loops drawn can be deformed as follows:

/2()/{1/3,2/3} ) = )/01 )/{1/3,2/3} YO'

/2()/~1/9 2/9)) -2 , = )/0 )/{7/9,8/9} )/~,

- - 1 - 1 2 fi/()/{4/9,519}) = )/0 )/{1/3,2/3} )/0 1 )/{4/9,5/9} )/{1/3,2/3} )/0'

/~()/{7/9, 8/9} ) m )/{-11/3,2/3 } )/01 )/{ 1/9, 2/9} )/01 )/{ 1/3, 2/3 } )/0"

It is easiest to find these formulas by noticing that the class of a loop is entirely described by the order in which it crosses the descending rays from critical points of H(r) , and the descending ray of argument 1/2. Notice that depending on the order of the crossing we might have to insert )/0 or )/o 1 (compare with/2(Ytl/9,2/9}) and/2()/{7/9,8/9}))"

From the expressions for/2 above it follows that

~()/~1/3 2/3))= -2 2 , )/0 )/(1/3,2/3} )/0,

/'/()/{1/9 2/9}) = )/0 2 -1 -1 3 , )/{1/3,2/3} )/{I/9,2/9} )/0 )/(1/3,2/3} )/0'

) / -2 -1 -1 - I -1 3 fl()/{4/9,5/9} ) = 0 )/{1/3,2/3})/0 )/{1/3,2/3} )/0 )/{4/9,5/9} )/{1/3,2/3} )/0)/{1/3,2/3} )/0'

//()/{7/9 8/9}) = )/O 1 -1 -1 2 , )/{1/3,2/3} )/0 )/{7/9, 8/9} )/{1/3, 2/3} )/0"

This way of computing rapidly gets out of hand. Perhaps there is a better notation for such expressions.

11.5. The fundamental group of the quotient parapattern bundle

We now want to compute the fundamental group of f2, which is obtained from ~ by the equivalence described in Section 7.6. More precisely, the points (~, ~3), (j~, ~3) and (j2~, ~3) are identified. This "identification locus" can be parametrized by the path

t ~ (re 2~it/3, r 3e2~#), 0 <~ t <~ 1,

i

B. BRANNER AND J. H. HUBBARD 316

UI

V2

Q(r) u Q(jr) u ~(j2r) Fig. 11.3

which is a loop in f~, starting at (r, r 3) and ending at (jr, r3). Let fl be the corresponding element of nl(fl, Po).

PROPOSITION 11.5. The group xl(ff2,Po) is freely generated by the image of xl((2,Po) and the element fl, subject to the relation fla=ct.

Proof. Let UI be a small neighborhood of fl; by choosing this neighborhood "tubular" we will have xl(U1,Po)~Z, generated by ft. Let V2 be the complement in of a smaller closed "tubular" neighborhood of ft. The open set V2 does not contain the base point; choose a path 6 in Q(r) connecting f~(r) I"1 V2 to Po and let U2 be the union of V2 and a small neighborhood of 6 (see Figure 11.3). Such a/-]2 is a deformation retract of ~, so the inclusion U2--~ is a homotopy equivalence.

Moreover, UI f3 U2 is a bundle of discs over C - / ) (with a handle, the part of the neighborhood of di in U1, attached). Thus nl(Ul N U2, Po) is canonically isomorphic to Z, and the inclusion U~ fl U2--~U2 maps the canonical generator to a. On the other hand, the inclusion U1 n U2--* U1 maps the generator to f13.

Now van Kampen's theorem tells us that nl(f2, Po) is the amalgamated sum of the diagram

Jrl(U2, Po)

"7'[l(U1 [7 U2, Po) :rl(f2, Po)

--... f n, (U,, Po)


which translates into

erl(Q, P0)

'7/ \ Z :rl(f2, Po)

Z

This is just another description of zq(Q, P0) by generators and relations; fl is the image of the generator of etl(U1). Q.E.D. for Proposition 11.5

11.6. The fundamental group of the escape locus

Since ~+ and ~ - are symmetrical, their fundamental groups are isomorphic; let fl-+, 7o + be the corresponding elements in zq(~+,P0). Note that fl+=fl-=fl. Van Kampen's theorem tells us that the fundamental group zq(~,Po) is the amalgamated sum of the diagram

f ~ l ( C ~ + , P o ) ~

~/'1(~ + f] ~ - ' Po) :rl(~, Po)

" ~ grl(C~-, po) f

So the only problem is to compute ffl~l(~+n ~-,P0) and the maps induced by the inclusions.

PROPOSITION 11.6. The group ~1(~ + N ~-,P0) is generated by 70 + and fl, subject to the relation [70+,fl3]= 1. The element 7o- satisfies y0 + fly0-=fl 2.

Proof. To keep the notation almost reasonable, denote T= ~+tq ~ - and let ATCT the polynomials with a single critical point, i.e. those Pa, b with a=O. Consider the mapping i~: T---~R+ x (R/Z) x (R/Z) defined by

f'(a, b) = (3 log s, arg(q0a, b(Pa, b(a))), arg(q0a, b(Pa, b(--a)))),

where logs=ha, b(+a)=ha, b(--a). This map is not quite a homeomorphism, it is a triple cover ramified along R+xA, where A is the diagonal of R/ZxR/Z as shown in Corollary 14.4 in [BH].

21t-928286 Acta Mathematica 169. Imprim6 le 10 novembre 1992


//,

/3

u

Fig. i 1.4

To find a lifting of F which is a homeomorphism, consider the subset S c R / Z x R/Z defined by S={(01, 02)1d(01, 02)~>1/3}, and the equivalence relation

(01, 02) = (0/ , 02') if 301 = 3 0 2 = 30 I' = 3 0 2 ' .

Set S=,~/~., and consider the map rt: ~r given by ~r(01, 02)=(301,302), which clearly induces a map S---~R/Z• still called ~r. Note that ~r is a triple cover ramified above A.

In the obvious way of representing the torus by a square with sides identified, the region S is as pictured in Figure 11.4. Moreover, the equivalence relation above identifies only points of the boundary, and identifies them if they have the same first or second coordinate. For future reference, denote u, o and w respectively the loops in S represented by the segments from (0, I/3) to (0, 2/3), (1/3, 0) to (2/3, 0) and (0, 1/3) to 0/3, 2/3) all of whose endpoints are identified in S to a single point P0. The other segments of the diagonal lines marked in the figure are identified with w in S.

If a~=0 then

~+(a, b) = limz__,_2aq~a,b(Z) and ~-(a, b) = limz_.,E~pa, b(z)

are well defined and and the mapping F: T-Ar---~R+x,~ given by

F(a, b) = (log s, arg(r b)), arg(r b)))

is an injective lifting off" on T -At . It does not extend to At, since there are then no co- critical points, but it does induce a mapping F: T---~R+ x S.

Since both the mapping


z#: T---> R+ x (R/Z)x (R/Z) and idxaz: R+ xS--> R+•215

are triple covers ramified over R+ x A, we see that F is a homeomorphism. Therefore

F , : :tl(T, P0)--> :tl(S, P0)

is an isomorphism. The homomorphism F , maps y0 + to u, Y0- to v and fl to w. From the description of S above, it is clear that u and w generate :rl(S, P0) and satisfy [u, w 3] = 1, and that u w v = w 2. Q.E.D. for Proposition 11.6

Remark 11.7. In s whose boundary is a torus, the lift of Y0- starting at the base point is not closed: its endpoints are identified by the projection (2-of2.

So a symmetrical description of the group :q(~e+N V-,P0) is: zq(~g+ n ~ - , P 0 ) i s generated by 3 elements ),o +, ~'o- and /3, satisfying 7o+fl~,o-=fl 2 a n d [~)0+,33] -'- D,o-,fl3] = 1.

COROLLARY 11.8. The group ~l(~,P0) /s the amalgamated sum o f the diagram

f ff/71 ( ~ +' P 0 ) , ~ .

~1(~ + n ~ - , Po) :rl(~, Po) - ... / "Tt'l ( ~ - ' Po)"

where the mappings zq(~ + n ~-,Po)---~:q(~+-,Po) map flE:rt(~ + n ~-,Po) to flEztl(~+-,Po) and yo • to yo~ztl(~•

Chapter 12. Polynomials of higher degree

Almost everything in this paper and the preceding one [BH] goes over to polynomials of arbitrary degree d, having two possibly multiple critical points. In this chapter we will outline how this goes, without proofs.

Our reason is not so much to generalize as to show the one thing which does not go through: Theorem 4.3 (a) and its consequences Theorem 5.2 and Theorem 9.1 (b) are false, or at least the proofs fail.



12.1. The global topology of parameter spaces

Choose d~ and d2 both positive with d l+d2- l=d , and set

P,,b(Z) = d (u- (d2-1)a) d'-I (u+(dl-1)a) e2-1 du+b

with critical points o)l=(d2-1)a and o )2=- (d l - 1)a, of local degree d l and d2 respectively. Clearly the case d1=d2=2 is the one we have been studying. We will say that such polynomials have bidegree (dl, d2).

We can define the connectedness locus

(~ dl, d 2 = { ( a, b ) E C2I the Julia set of P a, b is connected}

for this family, and the function H: C2--->R by

H(a, b) = sup{ha, b(o)l), ha, b(w2)}

which is continuous and vanishes exactly on cCd,,d2. The locus

Yr = {(a, b)l a(a, b)=log r}

is as before a 3-sphere for r> l , and H: C2- cCd,.d2--->R + is a trivial fibration. This shows that CCd~ ' d2 is cellular.

Remark 12.1. (a) Apparently our use of the word "cell-like" in [BH, Corollary 11.2] instead of cellular is inconsistent with previous usage [B]. Moreover, we claimed in the same corollary that the connectedness locus is contractible. One can only conclude weakly contractible.

(b) P. Lavaurs [L] has proved that the connectedness locus ~r is cellular in all degrees.

Further, we can write ~r-~-~rlu ,~r 2, where

5fr i = {(a, b) E ~rl H(a, b) = ha, b(o)i)} for i = 1,2.

Theorem 14.1 of [BH] then generalizes as follows: recall that ($3;E1, E2) denotes the standard decomposition of S 3 into two linked solid tori, with T=E~ N E2 canonically homeomorphic to S t x S l, and Ar the diagonal in T.

THEOREM 12.2. The triple (5er; 5r 1, 5e~ 2) is homeomorphic to the d-fold cover o f ($3; E 1, E 2) ramified along At.

POLYNOMIALS OF HIGHER DEGREE 321

12.2. Patterns in higher degrees

As in Chapter 2 we can construct ~-patterns of bidegree (dl, d2) for any dl, d2~>2. For each ~-pattern of depth N > 0 the mapping zR: R--~SN(R) wilt be a d-fold cover ramified at ~OR and with zeR of local degree dl at wR.

In the construction of patterns Lemma 2.2 was essential. The analogous lemma still holds with essentially the same proof:

LEMMA 12.3. Let a and b be two points in X=C. Then there exists a connected d- fold cover ~: Y---~X ramified above a and b such that z-I(a) and z~-J(b) each contain precisely one critical point, o f local degree dl and d2 points respectively. I f zcr: Y---~X and Ztz: Z---~X are two such covers, there exists a unique covering homeomorphism Y ~ Z .

With this lemma the complete construction of the pattern tree in Section 2.2 goes over exactly as in Chapter 2, except for the construction of iR: Ro-.-->R ~, which requires a small modification. In this case there is no unique co-critical point if d2>2, but it is possible to distinguish one of the d2-1 other (co-critical) inverse images of :rR-I((zIR (WR)) as follows.

The inverse image under xRt of the circle F0 of radius [~]d, which is contained in R0, is a graph FI in R1 consisting of d l - 1 loops which cover F0 with degree I and one loop AcaCI(R) which covers F0 with degree dE. All these loops touch at the point wR. The co-critical points all lie on A, hence are circularly ordered. Let us choose w[i to be the first one after WRI for this order.

Now analogously to Section 2.2 we can define iR: Ro---->R 1 as the unique covering isomorphisms R0- - -~ (R_I ) satisfying limz_~r iRI(Z)=WRI.

There are precisely d - 1 patterns isomorphic to a given one R E @(~); they lie in ~(~e z~i~(a-1)) where k=0, 1,..., d - 2 . The isomorphisms are constructed as in Section 2.7 by repeated liftings of the maps z~eZ~i/(d-~)Z, which commute with z ~ z d.

As in Section 2.8 we can construct pattern bundles. The adaptations of the statements in 2.8 are left to the reader.

All the statements in Chapter 3 go through in the obvious way.

12.3. Ends of patterns and tableaux

As in Chapter 4 for each R E ~ ( ~ ) we can construct tableaux satisfying precisely the same three tableau rules. Theorem 4.2 and Theorem 4.3 (b) and (c) are still true, but the


9 r ~ x

S S ) ' _ _ _ .

P I ' / I

2 3 5 8 ~ x A A A A - - A

J

/ /

/ /

/ /

/ /

/

/

/ /

/ /

/ /

/ /

/ /

/ /

/ /

/

/ /

2 -

Fig. 12.1. The Fibonacci grid.

13

statement corresponding to Theorem 4.3 (a)fails if d2>2. More precisely, in the proof of Lemma 4.5 we sum a geometric series with ratio 1/2; this ratio becomes l/d2 in general and the proof simply does not go through. In fact, the result is false, as shown in the example below.

Example 12.4. Consider the Fibonacci marked grid in Figure 12.1, which can be described more formally as the marked grid whose critical staircase has tips (bk, ak) where the ak are the Fibonacci numbers a~=2, a2=3, a3=5 ... . with ak+l=ak+ak_l and the bk satisfy the recursion relation bk+l=ak+bk starting with bl = 1.

This critical marked grid satisfies the tableau rules, and its originators are as marked: in particular there is exactly one in every row except the 0th which contains exactly 2. In this case the computation analogous to Lemma 4.5 gives

modN(c)= ~ mod(Cj)=mod(C0)+ --]--1 E m~ k(c)) j=o d 2-1 (t,k)

originator

=mod(C0)+ . 1 ~ , mod(C0)+ mod(Ci ) i

j=O

POLYNOMIALS OF HIGHER DEGREE 323

For d2>2 this gives the following formula for the modulus of the critical nest

d 2 mod N(c) = ~ mod(C0).

Hence the critical end is convergent and non-periodic.

Remark 12.5. There is another way of understanding this example. Recall from Theorem 4.3 (a) the definition of the function t. For this tableau, the set t-~(n) has exactly two elements for every n, so that for any k,

mod(C,) = 2 ~ , mod(Cj). gen(j)=k+ 1 d2 gen~=k

Therefore the series whose kth term is the sum of the moduli of generation k converges.

12.4. Julia sets of polynomials of higher degree

In light of the results above, we do not get the analogue of Theorem 5.2 in degree greater than 3. The best we can do without new techniques are the following results:

THEOREM 12.6. Let P be a polynomial o f bidegree (dl, d2). Suppose wl escapes to and d2=2. Then

(a) Kp is a Cantor set i f and only if the critical component o f Ke is not periodic, and (b) if Ke is a Cantor set, then it is o f measure O.

, THEOREM 12.7. Let P be a polynomial ofbidegree (dl, d2). I fw l escapes to ~ and the critical component o f Ke is non-recurrent, then Ke is a Cantor set o f measure O.

Proofs. The proof of Theorem 12.6 is exactly the same as for Theorem 5.2 and Theorem 5.9; for in that case Theorem 4.3 (a) does hold: the ratio of the geometric series occurring in Lemma 4.5 is 1/2.

As mentioned in the introduction to Chapter 5, there is an easier proof of Theorem 4.3 (a) in the case of non-recurrence: in that case the annuli of the critical nest have moduli which are bounded below. This still holds in higher degrees. Theorem 12.7 follows. Q.E.D. for Theorems 12.6 and 12.7

12.5. Parapatterns in higher degree

All the constructions and theorems in Chapter 7, 8, 9 go through to the case under consideration. Without repeating the statements, we want to isolate the following result, generalizing Remarks 8.3 and 9.4:

324 s . BRANNER AND J. H. HUBBARD

PROPOSITION 12.8. There exists a polynomial o f bidegree (dl, d2) realizing any pattern o f bidegree (d,, d2).

12.6. Ends o f p a r a p a t t e r n s a n d Julia sets

Theorem 5.2 proves the following folk-lore conjecture in degree 3.

CONJECTURE 12.9. Let P be a polynomial and let f2b be the set o f critical points with bounded orbits. The Julia set Je is a Cantor set if and only if for all to E f2b and for all n>0 P~ where Ke(to) is the component o f Ke containing co.

Fatou and Julia knew this when s and when ~b=f2. Furthermore, we know from the theory of polynomial-like mappings that if there exists a critical point to E f2b and an n>0 such that P~ then the Julia set Je is not a Cantor set. But we do not know in general whether the nonperiodicity condition of critical components of Ke is sufficient to guarantee that the Julia set is a Cantor set. We know that the proof given in this paper does not go through.

More specifically, by the analogue of Theorem 4.2 (a), the critical marked grid in Example 12.4 is the marked grid of some pattern of infinite depth, and by Proposition 12.8, there is a polynomial with this pattern. In fact, the careful reader will find going through the proof of Theorem 4.2 (a) with this particular critical marked grid in mind that there are infinitely many choices to make, hence infinitely many patterns and infinitely many polynomials of this sort.

For these polynomials, the critical component of Ke is recurrent and non-periodic; and the nest is convergent. We do not know whether this component is a point. By Sullivan's no wandering domains theorem [S], the component has empty interior, and in fact has measure zero (this last fact was pointed out to us by Curt McMullen).

Even though such a component has measure zero, and even if the critical component is a point so that the Julia set is a Cantor set, the Julia set might have positive measure. The proof of Theorem 5.9 does not extend to this case.

Furthermore, we know nothing about the component of ~(~) in the parameter space corresponding to such polynomials. Sullivan's theorem does not apply to parameter space, and as far as we know such a component might have interior, corresponding to some P-invariant Beltrami form carried by the Julia set (which would have to have positive measure for this to occur).

We cannot settle Conjecture 12.9, but feel that figuring out whether it is true for these examples is an interesting problem.

REFERENCES 325

For cubics the subset of ~ corresponding to polynomials with critical tableaux equal to the Fibonacci grid forms a fiber bundle over C - / ) with fibers a Cantor set. It can be shown to be connec ted and hence homeomorphic to the dyadic solenoid (Proposition 10.7 and Remark I0,10).

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(1918), 47-245. [L] LAVAURS, P., Thesis, Universit6 de Paris-Sud Centre d'Orsay, 1989. [MSS] MAIqE, R., SAD P. & SULLIVAN, D., On the dynamics of rational maps. Ann. Sci. Ecole

Norm. Sup. (4), 16 (1983), 193-217. [M1] MILNOR, J., Morse Theory. Princeton University Press, 1963. [M2] - - Remarks on iterated cubic maps. Preprint, 1986. [M3] - - Hyperbolic components in spaces of polynomial maps. Preprint, 1987. [S] SULLIVAN, D., Quasi-conformal homeomorphisms and dynamics. I: Solutions of the

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Received August 7, 1989 Received in revised form February 1, 1990

The iteration of cubic polynomials Part II: patterns and...

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